Renewal of Transient Spiral Modes in Disk Galaxies
S. Sridhar
TL;DR
This paper develops a nonlinear theory to analyze how transient spiral modes in disk galaxies leave resonant 'scars' that influence the regeneration of spiral structures, highlighting the role of past interactions in galaxy morphology.
Contribution
It introduces a nonlinear approach to resolve resonance singularities, quantifies the effects of scars on disk properties, and explains how scars facilitate the renewal of spiral modes.
Findings
Resonant scars are finite features around resonances with simple DF forms.
Scars cause shifts in disk properties like angular momentum and density.
Scars suppress damping of certain spiral modes, aiding their renewal.
Abstract
Spiral structure in disk galaxies could arise from transient modes that create conditions conducive for their regeneration; this is the proposal of Sellwood and Carlberg, based on simulations of stellar disks. The linear response of an axisymmetric stellar disk, to an adiabatic non-axisymmetric transient mode, gives a final distribution function (DF) that is equal to the initial DF everywhere in phase space, except at the Lindblad and corotation resonances where the final DF is singular. We use the nonlinear theory of adiabatic capture into resonance to resolve the singularities and calculate the finite changes in the DF. These take the form of axisymmetric "scars" concentrated around resonances, whose DFs have simple general forms. Global changes in the physical properties are explored for a cool Mestel disk: we calculate the DFs of scars, and estimate the changes in the disk angular…
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Renewal of Transient Spiral Modes in Disk Galaxies
S. Sridhar
Raman Research Institute, Sadashivanagar, Bangalore 560 080, India
Abstract
Spiral structure in disk galaxies could arise from transient modes that create conditions conducive for their regeneration; this is the proposal of Sellwood and Carlberg, based on simulations of stellar disks. The linear response of an axisymmetric stellar disk, to an adiabatic non-axisymmetric transient mode, gives a final distribution function (DF) that is equal to the initial DF everywhere in phase space, except at the Lindblad and corotation resonances where the final DF is singular. We use the nonlinear theory of adiabatic capture into resonance to resolve the singularities and calculate the finite changes in the DF. These take the form of axisymmetric “scars” concentrated around resonances, whose DFs have simple general forms. Global changes in the physical properties are explored for a cool Mestel disk: we calculate the DFs of scars, and estimate the changes in the disk angular momentum, surface density and orbital frequencies leading to shifts of resonances. Resonant torques between disk stars and any new linear non-axisymmetric mode are suppressed within a scar, as is epicyclic heating. Since all the resonances of a linear mode with the same angular wavenumber and pattern speed as its precursor lie inside the scars of the precursor, it suffers less damping. Hence, scars filter the spectrum of noise-generated modes, promoting the renewal of a few select modes. Relic scars sustained by a galaxy disk, due to past tidal interaction with a passing companion, may still be active enablers of non-axisymmetric modes, such as the two-armed “grand design” spiral patterns.
Galaxy: disk – Galaxy: kinematics and dynamics – solar neighborhood – galaxies: kinematics and dynamics – galaxies: spiral – galaxies: structure
††journal: ApJ
1 INTRODUCTION
The spiral structure of disk galaxies is presumably caused by density waves, whose origin and maintenance have been discussed for a long time; see e.g. the reviews Toomre (1977); Athanassoula (1984); Bertin & Lin (1996); Binney & Tremaine (2008); Sellwood (2014); Shu (2016). It has been argued that a spiral density wave could be a quasi-stationary mode of an underlying smooth axisymmetric disk of stars and gas (Lin & Shu, 1966; Bertin & Lin, 1996; Shu, 2016), but there is not much support for this notion from either linear perturbation theory or numerical simulations of initially axisymmetric stellar disks with a smooth distribution of stars in phase space (see Sellwood, 2014, and references therein). In a purely stellar disk, the wave decays due to the excitation of the epicyclic oscillations (“heat”) of disk stars, accompanying angular momentum transport across the disk. As the disk heats up, it becomes less efficient at sustaining the non-axisymmetric mode. When gas is also present, energy is dissipated through shocks in the perturbed gas flow. Even though the gas mass is smaller than the stellar mass, it is important for the secular evolution of the disk through new star formation.
A tightly-wound spiral density wavepacket in a stellar disk propagates inside/outside the corotation resonance (CR) radius with the inner/outer radial boundary located at the Inner and Outer Lindblad resonances (ILR/OLR), where it is absorbed as a short trailing wave (Toomre, 1969; Mark, 1974). A more global approach considers a spiral perturbation (a spiral “mode”) of fixed spatial form and pattern speed but with a time varying amplitude. Lynden-Bell & Kalnajs (1972) showed that a non-axisymmetric perturbation extracts angular momentum from the disk stars at the ILR and deposits it at the OLR, while heating up stars at both Lindblad resonances. There is no heating at the corotation resonance (CR) but angular momentum exchange can be of either sign, causing radial mixing of stellar orbits (Sellwood & Binney, 2002). A spiral density wave can undergo swing-amplification as it unwinds from a leading to a trailing one, but this is a transient phenomenon (Goldreich & Lynden-Bell, 1965; Julian & Toomre, 1966; Toomre, 1981). Global numerical linear theory computations by Toomre and Zang (Toomre, 1981) displayed striking swing amplification of an initial leading spiral wavepacket localized near the ILR. But this growth is transitory, and the wave unwinds until it is absorbed as a short, trailing wavepacket at the ILR. The “local” (e.g. Toomre, 1969) and “global” (Lynden-Bell & Kalnajs, 1972) linear theory calculations are complementary points of view of the same physical process. In the former the focus is on the conservation of the radial flux of angular momentum, which is the sum of the fluxes due to gravitational torques and advection, in the regions between the CR and ILR/OLR. In the latter the focus is on the total angular momentum budget, with lasting exchanges occurring only at the resonances.
Transient spiral patterns appear abundantly in numerical simulations of stellar disks (see e.g. Miller et al., 1970; Hockney & Brownrigg, 1974; James & Sellwood, 1978; Sellwood & Carlberg, 1984; Sellwood & Lin, 1989; Roškar et al., 2012), produced by swing-amplification of shot noise. They have the intriguing property that they can be reborn, a line of thought that goes back to Sellwood & Carlberg (1984); Sellwood (1989). As noted in Sellwood (2012), “ the decay of one spiral feature might change the background disk in such a manner as to create conditions for a new instability ”. The transient patterns are mode-like, with approximately constant spatial form and pattern speed but time-varying amplitudes; complex patterns seen in the simulations could be superpositions of a few spiral modes (Sellwood & Carlberg, 2014, hereafter SC14). The dynamical origins of this phenomenon, explored over decades of numerical simulations, is our central concern.
Plan of the paper. Spiral perturbations mainly affect in-plane stellar motions, so we focus on planar dynamics, as has much of the essential analytical and numerical work in this field cited above. In § 2 we discuss the decay of linear modes of a smooth axisymmetric DF from the point of view of Lynden-Bell & Kalnajs (1972). The emphasis is on the angular momentum transfer at the Lindblad and corotation resonances, with the associated excitation of epicyclic motions. In § 3 we take up the problem of the DF changes due to the passage of a single non-axisymmetric transient mode on an unperturbed, axisymmetric galactic disk. Linear response theory (Carlberg & Sellwood, 1985) predicts that, for an adiabatically varying transient, the change in the DF is singular at resonances. We use the nonlinear theory of adiabatic capture into resonance (Sridhar & Touma, 1996, hereafter ST96) to resolve the singularities. The post-transient (or final) DF is a simple function of the pre-transient (or initial) DF. The final DF is more “flattened” near a resonance when compared with the initial DF, because of mixing due to the separatrix crossing of stellar orbits; this has important consequences for mode renewal, as discussed in § 5 and 6. The DF of a “scar” is defined as the difference between the final and initial DFs. It is localized around resonances, and is the basic quantity required to calculate changes in all physical quantities. We derive an expression for the angular momentum absorbed from the transient by resonant stars.
The DF of scars are calculated in § 4 for a cool Mestel disk. This allows us to make quantitative estimates, guiding the rest of the paper. These DFs provide a phase space picture of mass shifts across resonant surfaces, which determine the angular momentum absorbed from the transient, and the surface density profiles of scars. Simple expressions are derived for both quantities, together with numerical estimates. In § 5 the results of the previous sections are gathered together, to prove that the linear responses of the initial and the final DFs are very different, to a new perturbation with the same angular wavenumber and pattern speed as the original transient mode. The flattening of the DF inside resonant scars suppresses resonant torques in the final disk.
In § 6 we address the problem of mode renewal in the final disk, by first showing that suppressed torques in the final disk also result in suppressed epicyclic heating. The suppression factor between the final and initial disks is small for linear modes with the same angular wavenumber and pattern speed as the transient. But it is likely to be of order unity for some other linear mode, whose resonant curves do not lie within the scars of the final disk; these modes will, generically, have different angular wavenumber or pattern speed. We propose a model of mode renewal, and compare with the simulations of SC14. Their physical insights regarding the growth, decay and renewal of non-axisymmetric modes generated by shot-noise are interpreted in terms of our model, wherein scars act as sharp filters of a noisy generator, favouring the growth of linear modes with the lowest dissipation rates. We conclude in § 7 with brief consideration of the limitations and desirable extensions of our model of mode renewal, possible numerical tests, prospects for seeing them in the second Gaia data release, and speculation on the role of relic scars on the evolution of galactic disks.
2 DECAY OF LINEAR MODES
2.1 Disk dynamics
The galactic disk consists of stars orbiting in a plane under the combined actions of their mutual self-gravity and external gravitational sources. Let be the polar coordinates of a star, and be the conjugate momenta; is the radial velocity and is the -component of the orbital angular momentum per unit mass. We use to denote phase space location and for phase volume. The mass of stars at time in is , where is the mass distribution function (DF) with . The total gravitational potential acting on a star is
[TABLE]
is the mean self-gravitational potential due to disk stars, where \mbox{\boldmathR}=(R,\phi) and \mbox{\boldmathR}^{\prime}=(R^{\prime},\phi^{\prime}) are position vectors in the disk plane. is the potential of a dark halo in the disk plane, and is a given external perturbation. The Hamiltonian governing the dynamics of a star is the energy per unit mass,
[TABLE]
The time evolution of the DF is governed by the collisionless Boltzmann equation (CBE):
[TABLE]
is the Poisson Bracket. Equations (1)–(3) determine the self-consistent evolution of the DF.
The unperturbed galactic disk is assumed to be stationary and axisymmetric, with Hamiltonian
[TABLE]
where . Both and are constant along a stellar orbit. The radial and angular frequencies of the orbit are functions of , and are generally incommensurate. So a generic orbit describes a “rosette”, as it goes through many periapse and apoapse passages, eventually filling an annular disk between the pericenter radius and the apocenter radius . Since the unperturbed DF is time-independent, Equation (3) and the Jeans theorem imply that it must be of the form . Let be a small perturbation to . In the limit of an infinitesimal perturbation, satisfies the linearized CBE (LCBE):
[TABLE]
where is the total potential perturbation.
2.2 Linear modes
We are interested in non-axisymmetric modal perturbations of the form,
[TABLE]
Here 0<\mbox{\varepsilon_{\rm p}}\ll 1 is a small parameter, is a constant pattern speed, and for a perturbation that is applied gradually in time; can be taken to zero at the end of the calculation.
Following Kalnajs (1971) we introduce the action-angle variables, , to study perturbations. The momenta are , where is the radial action. is a measure of the departure from a circular orbit: for a circular orbit, and the amplitude of radial excursions increases with increasing for given . The conjugate coordinates are , the radial and angular phases. The Hamiltonian governing the unperturbed time evolution of the action-angle variables is , the orbital energy per unit mass. This implies that and are constant along an orbit (which is expected), whereas their conjugate angles advance uniformly with time at the rates,
[TABLE]
We rewrite the unperturbed DF as , and expand the mode functions of Equation (6) as Fourier series in the angles:
[TABLE]
where the Fourier coefficients, \mbox{\widetilde{\Phi}}_{\ell m}=(\mbox{\widetilde{\Phi}}_{-\ell,-m})^{*} and \mbox{\widetilde{F}}_{\ell m}=(\mbox{\widetilde{F}}_{-\ell,-m})^{*}, are functions of , and the sums are over all integer pairs .
Using Equations (6) and (8) in the LCBE, we obtain the linear response as:
[TABLE]
where \omega=\left\{\ell\Omega_{R}+m\!\left(\Omega_{\phi}-\mbox{\Omega_{\rm p}}\right)\right\}. A stationary DF corresponds to a marginally growing mode, obtained in the limit :
[TABLE]
where is the Dirac delta-function. Both terms within the are singular at resonances in action space, where
[TABLE]
For given , an resonance is a curve in action space. Away from the resonance, the response of Equation (10) is non-singular and the -function does not contribute; mode structure may be studied by solving integral equations (see e.g. Kalnajs, 1971; Evans & Read, 1998). For a tightly-wound spiral density wave, the self-consistent problem becomes “local” and can be solved to obtain the well-known Lin-Shu-Kalnajs dispersion relationship (Lin & Shu, 1966; Kalnajs, 1965); see Binney (2013) for a nice derivation.
2.3 Angular Momentum budget
The -function part of the response of Equation (10) can be thought of as a “van Kampen” mode (see e.g. Binney & Tremaine, 2008). But it is a particular type of van Kampen mode, and plays a fundamental role in determining angular momentum exchanges between the mode and resonant stars (Lynden-Bell & Kalnajs, 1972). The total torque exerted by the mode on disk stars,
[TABLE]
is second order in the perturbation. Since the Poisson Bracket is invariant under canonical transformations, \mbox{\partial}\Phi_{1}/\mbox{\partial}\phi=[\Phi_{1},p_{\phi}]=[\Phi_{1},L_{z}]. Using Equations (6), (8) and (10) in Equation (12) — see § 2.4 of Kaur & Sridhar (2018) for a simple derivation in the context of an unperturbed spherical galaxy — we obtain the Lynden-Bell & Kalnajs (LBK) torque formula:
[TABLE]
The and resonances are the inner and outer Lindblad resonances, respectively; the principal ones are the ILR and the OLR , and is the corotation resonance (CR).
For the mode to be stationary, every ; for, if any , the exchange of angular momentum with resonant stars will make the perturbation time-dependent. Indeed, the as Lynden-Bell & Kalnajs (1972) demonstrated for “epicyclic” disks, which are reasonable first approximations to cool galactic disks in which radial speeds are much smaller than circular speeds. An epicyclic DF, , has (\mbox{\partial}F_{0}/\mbox{\partial}J_{R})<0 and |\mbox{\partial}F_{0}/\mbox{\partial}J_{R}|\gg|\mbox{\partial}F_{0}/\mbox{\partial}L_{z}|\,. Then \left(\ell\,\mbox{\partial}F_{0}/\mbox{\partial}J_{R}+m\,\mbox{\partial}F_{0}/\mbox{\partial}L_{z}\right)\simeq\ell\,\mbox{\partial}F_{0}/\mbox{\partial}J_{R} at all the Lindblad resonances, and =m\,\mbox{\partial}F_{0}/\mbox{\partial}L_{z} at the CR. Using this in Equation (13b) we can conclude:
a. at all the inner Lindblad resonances, and at all the outer Lindblad resonances.111In the complementary local approach, angular momentum is transported by wavepackets traversing the regions between the resonances, where the radial flux of angular momentum is the sum of gravitational and advective (“lorry transport”) fluxes. The wave angular density is negative/positive inside/outside corotation (Binney & Tremaine, 2008, § 6.2.6).
b. At the CR the sign of is opposite to that of \mbox{\partial}F_{0}/\mbox{\partial}L_{z}, so the angular momentum exchange can be of either sign.
If, instead of a marginally growing mode, one considered a growing mode with , then the -function in Equation (13b) would be replaced by a Lorentzian function of width . For not too large, the signs of the would not change. Hence, we may expect the conclusions of items (a) and (b) to be still valid. Moreover, the process is accompanied by the excitation of epicyclic motions at all the Lindblad resonances (see footnote 2). Therefore, we expect non-axisymmetric linear modes of a smooth DF to be generically transient.
3 RESONANT DEFORMATION DUE TO AN ADIABATIC TRANSIENT
What changes does an axisymmetric DF suffer due to a transient, non-axisymmetric mode? Carlberg & Sellwood (1985) calculated the changes to for a transient spiral mode whose amplitude grew as for and decayed as for . In the limit of slow growth and decay, , the DF shows no lasting changes anywhere in phase space, except at the resonances where the linear response is singular. The singularities can be resolved only through an intrinsically nonlinear treatment of the CBE of Equation (3). We calculate the finite changes in the DF near a resonance, using the nonlinear theory of adiabatic capture into resonance (ST96).
When there is not much overlap of resonances — which is indeed true for a cool Mestel disk; see § 4.2.1 and Figure 3 — the dynamics near a resonance can be reduced to the standard pendulum form (Chirikov, 1979; Monari et al., 2017), whose derivation is given in § 3.1. In § 3.2 we use the results of ST96 to determine the post-transient (or final) DF, which has the simple form given in Equation (27). The final DF is finite, and differs from the initial DF only inside a region of width O\!\left(\sqrt{\mbox{\varepsilon_{\rm p}}}\right) around a resonance. The changes in the disk are concentrated around resonances, which may be thought of as resonant scars left behind in phase space by the transient. In § 3.3 we define the DF of a scar, which is the basic physical quantity required for calculating changes in disk properties. Equation (30) for the DF of a scar shows that it varies by O\!\left(\sqrt{\mbox{\varepsilon_{\rm p}}}\right) over a region of width O\!\left(\sqrt{\mbox{\varepsilon_{\rm p}}}\right) in action space. We derive an expression for the angular momentum exchanged between the transient mode and resonant stars. In contrast to the change of the linear theory (see § 2.3), the change in the angular momentum within the scar is , which is larger. The global change in disk properties is explored for a cool Mestel disk in § 4 and 5.
3.1 Resonant dynamics
The unperturbed orbits of stars in action-angle space are governed by the Hamiltonian, , with orbital frequencies, and , given by Equation (7). The orbits are perturbed by a transient, adiabatically time-varying, non-axisymmetric mode with fixed angular wavenumber and pattern speed . The gravitational potential perturbation of the transient is:
[TABLE]
Here 0<\mbox{\varepsilon_{\rm p}}\ll 1 is a small, but not infinitesimal, measure of the perturbation strength; is a dimensionless time profile function, where is the ratio of the orbital periods of stars to the timescales over which varies; and are the radial profile and phase functions, respectively.
can be rewritten in terms of the action-angle variables, , as
[TABLE]
Here \,\mbox{\widetilde{\Phi}}_{\ell m}(J_{R},L_{z}) are complex Fourier coefficients that can be calculated from and . The perturbed orbits of stars are governed by the total Hamiltonian,
[TABLE]
Resonances occur on curves in the plane, on which Equation (11) is satisfied. We assume that the resonant curves for different , for the given , are well separated. For dynamics near some chosen resonant curve, the dominant contribution to comes from the term \bigl{[}\,\mbox{\widetilde{\Phi}}_{\ell m}\,\exp\left\{{\rm i}\left(\ell\theta_{R}+m(\theta_{\phi}-\mbox{\Omega_{\rm p}}t\right)\right\}\,+\,\mbox{c.c.}\,\bigl{]}.
The resulting resonant dynamics is conveniently described in terms of new canonical variables, which rotate with the perturbation at its pattern speed . These are the “fast” and “slow” actions and angles, (\mbox{J_{\rm f}},\mbox{J_{\rm s}};\mbox{\theta_{\rm f}},\mbox{\theta_{\rm s}}):
[TABLE]
In the co-rotating frame, every orbit acquires an extra regression of at the rate m\mbox{\Omega_{\rm p}}, so the unperturbed dynamics is described by the (Jacobi) Hamiltonian,
[TABLE]
The fast angle advances at the rate, \dot{\mbox{\theta_{\rm f}}}=\Omega_{\rm f}=\left(\mbox{\partial}H_{\rm r0}/\mbox{\partial}\mbox{J_{\rm f}}\right)=\Omega_{R}>0. The slow angle varies at the rate \dot{\mbox{\theta_{\rm s}}}=\Omega_{\rm s}, where the slow frequency is
[TABLE]
The resonance condition of Equation (11) is identical to \Omega_{\rm s}(\mbox{J_{\rm f}},\mbox{J_{\rm s}})=0, so is small near the resonance. Solving this, we can obtain the equation of the resonant curve in the (\mbox{J_{\rm f}},\mbox{J_{\rm s}}) plane as
[TABLE]
For galactic disks, we generally consider \mbox{J_{*}}>0.
The Hamiltonian governing dynamics near \mbox{J_{*}}(\mbox{J_{\rm f}}) is obtained by adding the potential perturbation, \bigl{[}\,\mbox{\widetilde{\Phi}}_{\ell m}\,\exp({\rm i}\mbox{\theta_{\rm s}})+\mbox{c.c.}\,\bigl{]}, to of Equation (18). We write \mbox{\widetilde{\Phi}}_{\ell m}=-(1/2)\mbox{\Phi_{\rm p}}\,\exp(-{\rm i}\xi), where \mbox{\Phi_{\rm p}}(\mbox{J_{\rm f}},\mbox{J_{\rm s}}) and \xi(\mbox{J_{\rm f}},\mbox{J_{\rm s}}) are real functions. Then the resonant Hamiltonian is:
[TABLE]
Since is independent of , \,\mbox{J_{\rm f}}=\mbox{constant} even though is time-dependent.222The constancy of the fast action implies that changes in and at the resonance are related to each other; (Sellwood & Binney, 2002). For smooth epicyclic DFs, items (a) and (b) of § 2.2 imply that and have the same sign, so at all the Lindblad resonances, and at the CR. Then the change in the epicyclic energy (per unit mass), , is positive at all the Lindlbad resonances, and very small at the CR. Thus the disk heats up. Resonant dynamics has been reduced to that of a system with 1.5 degrees-of-freedom, where (\mbox{J_{\rm s}},\mbox{\theta_{\rm s}}) dynamics is governed by , in which is treated as a constant parameter. Once \mbox{J_{\rm s}}(t) and \mbox{\theta_{\rm s}}(t) have been solved for, \mbox{\theta_{\rm f}}(t) can be determined by integrating \dot{\mbox{\theta_{\rm f}}}=\mbox{\partial}H_{\rm r}/\mbox{\partial}\mbox{J_{\rm f}}.
can be simplified further by expanding it in a Taylor series in about \mbox{J_{}}(\mbox{J_{\rm f}}). The first term, H_{\rm r0}(\mbox{J_{\rm f}},\mbox{J_{}}), can be dropped because it does not contribute to the dynamics of (\mbox{J_{\rm s}},\mbox{\theta_{\rm s}}). Since \left(\mbox{\partial}H_{\rm r0}/\mbox{\partial}J_{\rm s}\right)_{J_{\rm s}=J_{*}}=\Omega_{\rm s}(\mbox{J_{\rm s}},\mbox{J_{}})=0, the linear term in (\mbox{J_{\rm s}}-\mbox{J_{}}) is absent. Then the unperturbed Hamiltonian, H_{\rm r0}(\mbox{J_{\rm f}},\mbox{J_{\rm s}})\to(1/2)B_{*}(\mbox{J_{\rm s}}-\mbox{J_{}})^{2}, where B_{*}(\mbox{J_{\rm f}})=(\mbox{\partial}^{2}H_{\rm r0}/\mbox{\partial}J_{\rm s}^{2})_{J_{\rm s}=J_{*}}=(\mbox{\partial}\Omega_{\rm s}/\mbox{\partial}\mbox{J_{\rm s}})_{J_{\rm s}=J_{*}}. We can also set \mbox{J_{\rm s}}=\mbox{J_{}} in the perturbation: \,\mbox{\Phi_{\rm p}}(\mbox{J_{\rm f}},\mbox{J_{\rm s}})\to\Phi_{*}(\mbox{J_{\rm f}})=\mbox{\Phi_{\rm p}}(\mbox{J_{\rm f}},\mbox{J_{}}) and \,\xi(\mbox{J_{\rm f}},\mbox{J_{\rm s}})\to\xi_{*}(\mbox{J_{\rm f}})=\xi(\mbox{J_{\rm f}},\mbox{J_{}}). Then the resonant Hamiltonian governing (\mbox{J_{\rm s}},\mbox{\theta_{\rm s}}) dynamics reduces to the standard pendulum form:
[TABLE]
There are two small parameters in the problem: the nonlinearity of the perturbation, ; and the adiabaticity parameter, . Both are infinitesimal for the linear theory of quasi-steady perturbations discussed in § 2. Our goal is to take the next step, of dealing directly with the lowest-order nonlinear effects of a small, but not infinitesimal, . In this section we solve the problem for the adiabatic ordering, \varepsilon\ll\mbox{\varepsilon_{\rm p}}, the likely consequences of relaxing which are discussed briefly in § 7. The adiabatic dynamics of is controlled by the time-varying , which is conveniently kept track of by the “slow” time variable , so we write .
Figure 1 shows the phase flow of the “-frozen” of Equation (22) in (\mbox{J_{\rm s}},\mbox{\theta_{\rm s}}) space for some .333From Equation (17), we note that, as and increase by , increases by whereas increases by . So there are islands of which only one is shown in Figure 1. In this limit, is a constant of motion, so orbits follow its isocontours. The phase can always be chosen (by shifting by , if necessary) such that has the same sign as . Then the elliptic fixed point is at (\mbox{J_{*}},\xi_{*}), about which the entire Hamiltonian flow is reflection-symmetric in (\mbox{J_{\rm s}},\mbox{\theta_{\rm s}}) space. can be of either sign; we have chosen to represent the flow in Figure 1, because this is the case for the Mestel disk in the epicyclic limit — see Equation (44).444Were positive, the arrows in Figure 1 would be reversed, but there would be no change in the physics of the process discussed in this section. The upper and lower separatrices, and , are the two solutions of H=E_{\rm sx}(\tau;\mbox{J_{\rm f}})=\mbox{\varepsilon_{\rm p}}A(\tau)\Phi_{*}\leq 0, given by
[TABLE]
meet at the hyperbolic fixed point (\mbox{J_{*}},\xi_{*}\pm{\rm\pi}), and divide phase space into three different regions.
- a.
in regions I and III (Blue regions in Figure 1), which have circulating orbits whose advance monotonically. can be thought of as the limiting circulating orbits belonging to I and III, respectively. Since is an even function of (\mbox{J_{\rm s}}-\mbox{J_{*}}), a given corresponds to two orbits; one in I, and the other in III which is obtained by reflection about the elliptic fixed point.
- b.
in region II (Red region in Figure 1), which can be thought of as a resonant island. This consists of trapped orbits librating around the elliptic fixed point, where takes its maximum value of .
- c.
In II, the libration frequency decreases as decreases, from at the elliptic fixed point for , to zero as . As decreases further, we enter I and III where the circulation frequency is zero for and increases in magnitude as decreases further.
When the variation of with is taken into account, is no longer a conserved quantity. Since \varepsilon\ll\mbox{\varepsilon_{\rm p}}, the motion is adiabatic in regions that are not in the immediate vicinity of . Then the phase area, \oint{\rm d}\mbox{\theta_{\rm s}}(\mbox{J_{\rm s}}-\mbox{J_{*}}) enclosed by isocontours of , is an adiabatic invariant. This is a function of which is discontinuous across , because the topology of orbits changes from trapped to circulating or vice versa. The fate of an orbit when it encounters a slowly moving separatrix is very sensitive to at encounter, and the phase area is no longer a conserved quantity. For instance, a circulating orbit could get captured by the resonant island, or become another circulating orbit on the other side of the island; or a trapped orbit may escape from the resonant island through either of the separatrices. This process was described in probabilistic terms by Goldreich & Peale (1966) for single orbits in the context of solar system dynamics. ST96 applied this to stellar systems, and derived the equations governing the time evolution of the DF. Below we apply ST96 to derive the final DF, in terms of the above initial DF, after the passage of an adiabatic transient.
3.2 Post-Transient DF
For a transient mode, we assume that in Equation (22) increases monotonically from zero at , reaches a maximum value of unity at , and then decreases monotonically to zero at . At when is zero, the initial DF,
[TABLE]
is given as a smooth function of (\mbox{J_{\rm f}},\mbox{J_{\rm s}}). Here we derive the final DF after the passage of the transient.
As noted in the final paragraph of § 3.1, adiabatic dynamics conserves the phase area, \oint{\rm d}\mbox{\theta_{\rm s}}(\mbox{J_{\rm s}}-\mbox{J_{}}), of orbits that are away from . It is a function of , with discontinuities across . ST96 uses the conservation and non-conservation of the phase area in a basic manner. Since is an even function of (\mbox{J_{\rm s}}-\mbox{J_{}}), so is the phase area. This means — see item a of § 3.1 — that a given value of the phase area refers two different orbits, one in I and its reflection in III. But the initial DF of Equation (24) is, in general, not reflection-symmetric about \mbox{J_{\rm s}}=\mbox{J_{*}}; see Equation (48) for the Mestel disk. So the phase area is not a good variable to use to follow the evolution of the DF. However, this is readily fixed by exploiting a certain latitude in the choice of the relative signs in the three regions. Accordingly, we define
[TABLE]
which is an adiabatic invariant for orbits away from . The integrals are taken over the -frozen orbits, (\mbox{J_{\rm s}}-\mbox{J_{}})=\pm\sqrt{2[H+\mbox{\varepsilon_{\rm p}}A(\tau)\Phi_{*}\cos(\mbox{\theta_{\rm s}}-\xi_{*})]/B_{*}}, as applicable. Unlike the phase area, takes different values in I and III. Since \dot{\mbox{\theta_{\rm s}}}=B_{*}(\mbox{J_{\rm s}}-\mbox{J_{}}), we have \oint{\rm d}\mbox{\theta_{\rm s}}(\mbox{J_{\rm s}}-\mbox{J_{}})=B_{*}\oint{\rm d}t\,(\mbox{J_{\rm s}}-\mbox{J_{}})^{2}\leq 0. So in I, and in II and III: indeed, two orbits in I and III with the same have equal and opposite values of . So is a good variable to use to describe the evolution of a general initial DF of Equation (24).555A similar choice is implicit in the ranges given in Equation (6) of ST96, but was not stated explicitly there.
ST96 provides the equations governing the adiabatic time evolution of the DF as a function of (\mbox{J_{\rm f}},K). These are used in Appendix A to derive Equation (27) for the final DF in seven simple steps. Below we give a brief account of the physics of this process.
The first step is to write the initial DF as a function of and . At the initial time , when , the disk is axisymmetric with DF given by Equation (24). Both collapse to the line \mbox{J_{\rm s}}=\mbox{L_{}}, so trapped orbits are absent. Since the measure of region II is zero, region I consists of orbits with \mbox{J_{\rm s}}\leq\mbox{J_{}} and region III has orbits with \mbox{J_{\rm s}}\geq\mbox{J_{}}. Circulating orbits are governed by H=(1/2)B_{*}(\mbox{J_{\rm s}}-\mbox{L_{}})^{2}. Then \mbox{J_{\rm s}}=\mbox{constant} along orbits, so (\mbox{J_{\rm s}}-\mbox{J_{}}) can be pulled out of the integral in Equation (25). Since \dot{\mbox{\theta_{\rm s}}}=B_{*}(\mbox{J_{\rm s}}-\mbox{J_{}}) is positive in I and negative in III, \,\oint{\rm d}\mbox{\theta_{\rm s}} is equal to in I and in III. Hence K=\mbox{J_{\rm s}}-\mbox{J_{}}, and the initial DF can be written as \mbox{F_{\rm in}}(\mbox{J_{\rm f}},\mbox{J_{}}+K). Identical considerations apply at the final time , when again.
Between and the transient is non-zero. As expand and contract symmetrically about the resonant line \mbox{J_{\rm s}}=\mbox{J_{*}}(\mbox{J_{\rm f}}), the disk undergoes a sequence of adiabatic transformations through non-axisymmetric states. But the final DF is axisymmetric, just like the initial DF. Orbits may be divided into two classes; those that never cross , and those that experience separatrix-crossing. The dividing line between these two classes is given by the maximum excursions of away from , which happens at when . At this time, the corresponding to are equal to \pm\Delta J(\mbox{J_{\rm f}}), where
[TABLE]
is derived in Equation (A8).
1. Orbits with |\mbox{J_{\rm s}}-\mbox{J_{}}|\geq\Delta J at the initial time never encounter . These remain circulating orbits for all time, and is an adiabatic invariant. So the DF maintains a frozen form when expressed as a function of (\mbox{J_{\rm f}},K). Since K=\mbox{J_{\rm s}}-\mbox{J_{}} at the initial and final times, the final DF is equal to the initial DF.
2. Orbits with |\mbox{J_{\rm s}}-\mbox{J_{*}}|<\Delta J at the initial time experience separatrix crossing twice, first in the expansion phase and next in the contraction phase. Consider the circulating orbits with in region III and their reflected counterparts with in region I, where can take any value between [math] and .
2a. Let be the time between and when the expanding cross , respectively. The circulating orbits are now trapped by the expanding island, and turned into librating orbits with , just inside . Since the motions of the separatrices are reflection-symmetric, according to ST96 the DF of the newly formed librating orbits contains equal mixtures of the DFs of the circulating orbits at .
2b. Following this, expand away from these librating orbits, attaining maximum expansion at , after which they contract. Meanwhile, the DF of the librating orbits at remains frozen until encounter them during their contraction phase at the unique time (which lies between and ).
2c. As cross them, the librating orbits are liberated from the shrinking island into circulating orbits with in III and I, respectively. According to ST96, the DFs of these circulating orbits are equal to the DF of the librating orbits at just prior to the encounter.
2d. Further contraction of leaves the DF of unchanged until the final time . Hence the final DF at (\mbox{J_{\rm f}},\mbox{J_{\rm s}}) is a mixture containing equal proportions of the initial DF at (\mbox{J_{\rm f}},\mbox{J_{\rm s}}) and the initial DF at (\mbox{J_{\rm f}},2\mbox{J_{*}}-\mbox{J_{\rm s}}).
Therefore, the final axisymmetric DF is:
[TABLE]
As shown in Figure 2, the final DF is equal to the initial DF, except within a “scar” of half-width \Delta J(\mbox{J_{\rm f}})\propto\sqrt{\mbox{\varepsilon_{\rm p}}} straddling the resonant line. \mbox{F_{\rm fin}}(\mbox{J_{\rm f}},\mbox{J_{\rm s}}) is discontinuous at the scar boundaries, \mbox{J_{\rm s}}=\mbox{J_{}}(\mbox{J_{\rm f}})\,\pm\,\Delta J(\mbox{J_{\rm f}}), because there is a sharp difference, in the adiabatic limit, between the dynamical histories of regions that have experienced separatrix crossing and those that have not. Within the scar, is an even function of (\mbox{J_{\rm s}}-\mbox{J_{}}).
We can simplify Equation (27) by using the fact that \mbox{F_{\rm in}}(\mbox{J_{\rm f}},\mbox{J_{\rm s}}) is a smooth function of , and the scar occupies a narrow, O(\sqrt{\mbox{\varepsilon_{\rm p}}}), region of phase space. So, for |\mbox{J_{\rm s}}-\mbox{J_{}}|<\Delta J, the DF can be expanded in a Taylor series about \mbox{J_{\rm s}}=\mbox{J_{}}(\mbox{J_{\rm f}}):
[TABLE]
where F^{(n)}_{\rm in}(\mbox{J_{\rm f}})=(\mbox{\partial}^{n}F_{\rm in}/\mbox{\partial}J^{n}_{\rm s}) evaluated at \mbox{J_{\rm s}}=\mbox{J_{*}}(\mbox{J_{\rm f}}) are functions of . Using this in Equation (27), we obtain
[TABLE]
Within the scar, the final DF is a sum over the even terms of Equation (28). Comparing Equation (28) with (29) we see that, well within the scarred region, is a linear function of (\mbox{J_{\rm s}}-\mbox{J_{}}), whereas is a quadratic function of (\mbox{J_{\rm s}}-\mbox{J_{}}). Close to the resonance, the final DF is a more flattened function of (on any \mbox{J_{\rm f}}=\mbox{constant} surface) than the initial DF; see Figure 2. This flattening is analogous to the resonant flattening of the velocity space DF in the Landau damping of plasma waves.
3.3 DF of a scar
Since the final DF differs from the initial DF only within a narrow strip around the resonant line, it is useful to isolate these changes by defining the DF of a scar as the difference between the final and initial DFs:
[TABLE]
This is indicated in Figure 2 by the shaded regions between the final and initial DFs. Within the scar, is an odd function of (\mbox{J_{\rm s}}-\mbox{J_{}}), and discontinuous at the scar boundaries. Since \int{\rm d}\mbox{J_{\rm s}}\,\Delta F(\mbox{J_{\rm f}},\mbox{J_{\rm s}})=0\,, the total mass in the scar is zero. At each value of , mass has been shifted across the resonance. In the example shown in Figure 2, is positive for \mbox{J_{\rm s}}>\mbox{J_{}} and negative for \mbox{J_{\rm s}}<\mbox{J_{*}}.
The DF of a scar is the basic quantity required to calculate changes in the physical properties of the disk. is used in § 3.3.1 to derive a formula for the angular momentum absorbed from the transient mode by resonant stars. Below we describe some general features of the geometry of a scarred region in the (\mbox{J_{\rm f}},\mbox{J_{\rm s}}) plane, and then simplify the expression for .
a. We can simplify Equation (30) by using the Taylor series expansion of Equation (28). Then
[TABLE]
Within the scar, is equal to times the sum of the odd terms of Equation (28). The first term is O(\sqrt{\mbox{\varepsilon_{\rm p}}}), whereas the second term is , which is much smaller. Hence has O(\sqrt{\mbox{\varepsilon_{\rm p}}}) variations across the scar of width \propto\sqrt{\mbox{\varepsilon_{\rm p}}}. Therefore, a good measure of the nonlinearity of the problem is given by the ratio,
[TABLE]
From Equation (31), we can also infer that the direction of the shift of mass across the resonance depends mainly on the sign of F^{(1)}_{\rm in}(\mbox{J_{\rm f}}). Generally, for , mass has been shifted from (\mbox{J_{\rm s}}\gtrless\mbox{J_{}}) to (\mbox{J_{\rm s}}\lessgtr\mbox{J_{}}).
b. The extent of the scar along can be determined as follows: Since , Equation (17) implies that \mbox{J_{\rm f}}\geq-\ell\mbox{J_{\rm s}}. Then \mbox{J_{\rm f}}\geq j_{0}, where is the value of where the resonant line \mbox{J_{\rm s}}=\mbox{J_{*}}(\mbox{J_{\rm f}}) intersects the lower boundary \mbox{J_{\rm f}}=-\ell\mbox{J_{\rm s}}. Hence the minimum value of satisfies the equation
[TABLE]
For the CR, because . For the Lindblad resonances , and this equation must be solved to get . Since is usually positive for galactic disks, for the inner Lindblad resonances (which have ) and for the outer Lindblad resonances (which have ).
c. The half-width of the scar \Delta J(\mbox{J_{\rm f}}) is determined by the radial profile of the transient mode, and can assume varied forms depending on whether the perturbation is spiral or bar-like. The functional form must be either calculated from a model of observational data, or taken from numerical simulations. One general property we can infer directly is that it must vanish at the lower limit of , i.e. \,\Delta J(\mbox{J_{\rm f}})\to 0\, as \,\mbox{J_{\rm f}}\to j_{0}, at all the Lindblad resonances.666This can be understood as follows. At \mbox{J_{\rm f}}=j_{0}, the allowed values of are one-sided about the resonant value, \mbox{J_{}}(j_{0})=-j_{0}/\ell>0; for the inner Lindlblad resonances \mbox{J_{\rm s}}\leq\mbox{J_{}}(j_{0}), and for the outer Lindblad resonances \mbox{J_{\rm s}}\geq\mbox{J_{}}(j_{0}). But, for the pendulum Hamiltonian of Equation (22), the allowed deviations of must necessarily be symmetric about \mbox{J_{}}(j_{0}). This is possible only when both collapse to the line \mbox{J_{\rm s}}=\mbox{J_{}}(j_{0}), when the resonant island has zero area. Therefore at all the Lindblad resonances. We saw in item (b) above that for the CR, so symmetrical deviations of about \mbox{J_{}}(j_{0}=0) are indeed allowed, and need not vanish at the CR.
3.3.1 Angular Momentum exchange
The transfer of mass across the resonant line leads to a change in the angular momentum of stars in the scarred region. The angular momentum absorbed by the resonant stars from the transient mode is:
[TABLE]
where we have used \int{\rm d}\mbox{J_{\rm s}}\,\Delta F(\mbox{J_{\rm f}},\mbox{J_{\rm s}})=0\,. Using Equation (31), we obtain
[TABLE]
This shows that is generically , which is somewhat larger than the change of the linear theory discussed in § 2.3. The sign of can be positive or negative. This is computed in § 4.3, where it will become clear how differently physical quantities behave at the three kinds of resonances, namely the inner and outer Lindblad resonances and the CR.
4 SCARS IN A COOL MESTEL DISK
Equations (30) and (31) are compact expressions for , the DF of the axisymmetric scar left behind by the passage of a transient, adiabatic, non-axisymmetric mode. But these hide the important dependence of the sense of mass shifts across resonances — hence the signs of the resonant angular momentum exchanges, of Equation (35) — on the integers labelling different resonances. Only when this is revealed would we have a physical picture of the interactions of the transient non-axisymmetric mode with stars at the inner/outer Lindblad and corotation resonances.
In this section, we make explicit the general results of § 3, and compute physical quantities for a cool Mestel disk, a model that has been used extensively in analytical and numerical work. Some properties of a cool Mestel disk are summarized in § 4.1; see Binney & Tremaine (2008) and references therein for more details. The effect of the transient mode on the disk is considered in § 4.2, where we compute resonance locations and scar widths. The principal resonances, ILR (), the CR () and the OLR (), are well-separated in phase space. In § 4.3 we compute and discuss its properties as a function of , paying attention to the principal resonances.
The rearrangement of mass in phase space is best appreciated by plotting in the plane. This is because (\mbox{J_{\rm f}},\mbox{J_{\rm s}}) are defined only in the neighborhood of resonances, whereas are global coordinates; we always have , and for a disk of prograde stars. In § 4.4 we discuss the properties of in the plane, and its direct implications for the mass shifts at the principal resonances. Descending from phase space to real space, induces an axisymmetric change, , in the disk surface density. This is computed in § 4.5 and Appendix B.1, and found to be localized around resonant radii. The gravitational perturbation due to is computed in § 5, and used to showed that resonant torques in the scarred Mestel disk, due to a certain class of linear modes, are highly suppressed.
4.1 Unperturbed disk
The exact DF for a Mestel disk, given in Toomre (1977); Binney & Tremaine (2008), consists of stars orbiting only in a prograde sense, so . The surface density, , gives rise to an attractive, radial gravitational force , so the circular speed of a star is independent of (i.e. a flat rotation curve). The disk is embedded in a static spherical halo with three dimensional mass density profile , which exerts a radial force . The total radial acceleration felt by a star, due to disk and halo, in the disk plane can be written as , where is the constant circular speed. We write , where is the fractional contribution of the disk to the total radial force. The radial velocity dispersion is constant.
In a cool disk , and most of the stars are on near-circular orbits. The guiding-center radius of an orbit, , is determined by solving , which gives . The orbital and epicyclic frequencies are given by
[TABLE]
The typical epicyclic radius is , where is a dimensionless measure of disk coolness. In the epicyclic limit, , the radial action is given by
[TABLE]
The Schwarzschild DF for a cool Mestel disk is
[TABLE]
where is a constant. At any given , most stars have . Figure 3 shows some isocontours of in the plane and the locations of the principal resonances.777The loci of the resonances are straight lines of infinite slopes in the epicyclic limit we use. The resonant curves actually have finite slopes, but this does not affect our calculations because we calculate in the epicyclic limit, .
The epicyclic Hamiltonian governing the unperturbed stellar orbits is the sum of the energies (per unit mass) in circular and epicyclic motions:
[TABLE]
where is an arbitrary length scale. Then the radial and angular frequencies are
[TABLE]
where the term in is usually dropped in epicyclic theory.
4.1.1 Stability, responsiveness and choice of parameters
For the disk to be stable to axisymmetric perturbations, the Toomre parameter, . In a Mestel disk, is a constant independent of . Since , we must have . Since , this is always satisfied if . For fiducial values, and , we have , so the disks we consider are indeed linearly stable to axisymmetric perturbations. Then the relative magnitude of the term that was dropped in Equation (40b) is . The ratio of the typical epicyclic radius to the guiding-center radius is, . Whereas the unperturbed disk must necessarily be linearly stable, it must also be responsive to non-axisymmetric perturbations. This implies that should not be too large; generally, for the disk to be responsive to swing amplification of non-axisymmetric perturbations (Binney & Tremaine, 2008, § 6.3).
Besides , which sets the velocity scale, the natural parameters of the cool Mestel disk are the dimensionless quantities, and . This is because they can be simply and independently specified: is tightly constrained by axisymmetric stability and responsiveness to non-axisymmetric perturbations; we need only require , but we will use , which is more relevant to galaxies like the Milky Way. In the calculations below we use and , because the formulae appear simplest in these variables. When specific estimates are needed, we set , and express quantities in terms of and .
4.2 Perturbation due to a transient mode
The disk is perturbed by an -armed, small-amplitude, adiabatic transient mode with pattern speed , as discussed in the previous section. The potential perturbation is given by Equations (14) and (15), and reproduced below:
[TABLE]
The transient leaves in its wake scars at different resonances spread across the disk. Here we compute resonance locations and scar widths.
4.2.1 Resonance locations and unperturbed frequencies
Using the epicyclic frequencies of Equations (40a) and (40b) in Equation (11), we obtain the resonance condition,
[TABLE]
Solving this, the resonant line in the plane as L_{z}=\mbox{L_{*}}=\left(V_{0}^{2}/\mbox{\Omega_{\rm p}}\right)\!\left[1+\sqrt{2}\,(\ell/m)\right], which is independent of . Since the Mestel DF consists of stars with positive orbital angular momenta, the resonance exists only when ; henceforth we assume that are such that .888For an mode this means that , so there are no inner (i.e. ) Lindblad resonances; for an mode , so the Inner (i.e. ) Lindblad resonance exists but not the higher-order inner (i.e. ) Lindblad resonances; and so on for higher modes.
We can introduce the fast and slow action-angle variables of Equation (17) in the vicinity of the resonant lines, , in the plane. In the (\mbox{J_{\rm f}},\mbox{J_{\rm s}}) plane, the equation for the resonant line is \mbox{J_{\rm s}}=\mbox{J_{*}}, where
[TABLE]
is independent of . The principal resonances are indeed well-separated in action space (see Figure 3), as was assumed in the derivation of the pendulum Hamiltonian in § 3.1. The fast and slow frequencies can be calculated from Equations (40a) and (40b). \Omega_{\rm f}(\mbox{J_{\rm s}})=\Omega_{R}=\sqrt{2}V_{0}^{2}/m\mbox{J_{\rm s}}\sim O(1), and
[TABLE]
Both frequencies are independent of in the epicyclic limit. \Omega_{\rm s}(\mbox{J_{\rm s}}) has a resonance-independent form when expressed in terms of (\mbox{J_{\rm s}}/\mbox{J_{}}), and goes through zero at resonance, \mbox{J_{\rm s}}=\mbox{J_{}}.
4.2.2 Post-transient scar widths
The resonant dynamics during the passage of the transient is described by the pendulum Hamiltonian of Equation (22), which is reproduced below:
[TABLE]
Differentiating Equation (43), we find
[TABLE]
is independent of in the epicyclic limit. When the potential perturbation arises mainly from the self-gravity of the perturbation to the disk surface density, then \mbox{\varepsilon_{\rm p}}=\eta\,\mbox{\varepsilon_{\rm d}}, where is the fractional surface density perturbation (which is more convenient to specify). is the Fourier coefficient of the normalized potential perturbation. We write it as \Phi_{*}(\mbox{J_{\rm f}})=-({\rm\pi}V_{0}/4)^{2}\,D_{\ell m}(\mbox{J_{\rm f}}), where D_{\ell m}(\mbox{J_{\rm f}}) is a dimensionless, positive function.
As discussed in the previous section, the post-transient disk is also axisymmetric. The final DF is equal to the initial DF everywhere in phase space, except in narrow regions straddling different resonant lines. The half-widths of the scarred regions, given in Equation (26), are:
[TABLE]
At a given value of in the (\mbox{J_{\rm f}},\mbox{J_{\rm s}}) plane, the scar extends over a narrow strip, |\mbox{J_{\rm s}}-\mbox{J_{}}|<\Delta J(\mbox{J_{\rm f}}). The allowed values of can be determined as follows. The minimum value of the fast action, , satisfies Equation (33). Since the of Equation (42) is independent of , the solution to Equation (33) is just j_{0}=-\ell\mbox{J_{}}. Hence -\ell\mbox{J_{*}}\,\leq\,\mbox{J_{\rm f}}.
From Equation (32), the nonlinearity of the scar at the resonance is
[TABLE]
The D_{\ell m}(\mbox{J_{\rm f}}) diminish rapidly as increases, so the important resonances have small . Our normalization is such that D_{0m}(\mbox{J_{\rm f}}) is a function. But D_{-1m}(\mbox{J_{\rm f}}) and D_{1m}(\mbox{J_{\rm f}}) can be expected to be smaller because spiral density waves have the largest amplitudes near the CR and only extend about as far as the ILR or OLR; see e.g. Figures 5 and 6 of SC14. We make an estimate of scar widths at the principal resonances by setting , . For a Mestel disk with , and an transient with \mbox{\varepsilon_{\rm d}}=10^{-2}, we have
[TABLE]
at the ILR, CR and OLR, respectively.
4.3 Angular Momentum transfer
We use Equation (35) to estimate the angular momentum absorbed by resonant stars from the transient. The initial DF near the resonance is \,\mbox{F_{\rm in}}(\mbox{J_{\rm f}},\mbox{J_{\rm s}})\,=\,F_{0}(\mbox{J_{\rm f}}+\ell\mbox{J_{\rm s}}\,,\,m\mbox{J_{\rm s}}), where is the Mestel DF of Equation (38). This gives
[TABLE]
Differentiating with respect to , and setting \mbox{J_{\rm s}}=\mbox{J_{*}}, we have
[TABLE]
Substituting Equations (49) and (45) in (35),
[TABLE]
The functions D_{\ell m}(\mbox{J_{\rm f}}) determine the precise shapes of the scar boundaries in phase space. As discussed in item (c) following Equation (30), they are completely determined by the radial profile of the transient mode. D_{\ell m}(\mbox{J_{\rm f}}) can assume varied functional forms, depending on whether the transient is of spiral form or bar-like. The functional forms must be either calculated in a model of observational data, or taken from simulations. This is beyond the scope of this paper; we move forward by replacing D_{\ell m}(\mbox{J_{\rm f}}) by an effective resonance-dependent constant, . Then the -integral can be evaluated, and we obtain
[TABLE]
where is the total orbital angular momentum in stars of the unperturbed disk with L_{z}\leq L_{\rm CR}=V_{0}^{2}/\mbox{\Omega_{\rm p}}, and
[TABLE]
is a resonance-dependent angular momentum absorption coefficient.
Since has the same sign as , we have at all the inner Lindblad resonances, at the corotation resonance, and at all the outer Lindblad resonances. From footnote 2, the change in the epicyclic energies (per unit mass) of resonant stars (“heating”) is . There is no heating at the CR, whereas at all the Lindblad resonances. Over its passage, the transient non-axisymmetric mode has transferred angular momentum from the inner to the outer Lindblad resonances while heating up stars, which is in the same sense as the linear theory of Lynden-Bell & Kalnajs (1972) discussed in § 2.3. But , which is larger than the change expected from the linear theory.
4.4 Scars in phase space and real space
The angular momentum exchanges described above are a result of mass transfer across resonant surfaces. As discussed in § 3.3, the basic quantity describing these changes in phase space is the DF of a scar, \Delta F(\mbox{J_{\rm f}},\mbox{J_{\rm s}}), given by Equation (30) or (31). In § 4.4.1 we derive an explicit expression for at any resonance in a cool Mestel disk. The global rearrangement of mass in phase space is best appreciated when is plotted as a function of the global coordinates . This is done in Figure 4 by assuming that the scar width is a constant, as was used to derive Equation (51) for . In § 4.4.2 we integrate to obtain , the axisymmetric perturbation to the disk surface density.
4.4.1 Global changes in the DF
The DF of the scar, \Delta F(\mbox{J_{\rm f}},\mbox{J_{\rm s}}), at the resonance of a cool Mestel disk can be obtained by using Equation (49) in (31). is zero for |\mbox{J_{\rm s}}-\mbox{J_{}}|\geq\Delta J(\mbox{J_{\rm f}}), where and \Delta J(\mbox{J_{\rm f}})\, are given in Equations (42) and (45). For |\mbox{J_{\rm s}}-\mbox{J_{}}|<\Delta J(\mbox{J_{\rm f}}), the DF of the scar is
[TABLE]
Within the scarred region, is an odd function of (\mbox{J_{\rm s}}-\mbox{J_{}}), so the mass in every scar is zero. The equation of the resonant line in the plane is L_{z}=\mbox{L_{}}, where
[TABLE]
The half-width of the scar, \Delta L=m\,\Delta J(\mbox{J_{\rm f}}), depends on because \mbox{J_{\rm f}}=J_{R}-(\ell/m)L_{z}\,:
[TABLE]
We write the DF of the scar, \Delta F(\mbox{J_{\rm f}},\mbox{J_{\rm s}})=\Delta F_{\ell m}(J_{R},L_{z}), to emphasize its dependence on . vanishes for |L_{z}-\mbox{L_{}}|\geq\Delta L. For |L_{z}-\mbox{L_{}}|<\Delta L, it can be written as a function of by substituting \,\mbox{J_{\rm f}}=J_{R}-(\ell/m)L_{z}\, and \,\mbox{J_{\rm s}}=L_{z}/m\, in Equation (53):
[TABLE]
We still need to specify a functional form for D_{\ell m}(\mbox{J_{\rm f}}). As discussed in § 4.3, this requires information on the radial profile of the transient mode. We made the simplifying assumption D_{\ell m}(\mbox{J_{\rm f}})\to\bar{D}_{\ell m}, and derived Equation (51) for . The discussion following this equation provides some assurance that the resulting properties of are consistent with what may be expected of angular momentum exchanges between a non-axisymmetric mode and resonant stars. But it is a poor representation of the shape of a scar. Henceforth we use
[TABLE]
which is useful in providing a global picture of resonant mass shifts and in calculating the perturbation to the surface density. In Figure 4 we plot near the principal resonances of an transient. The specific functional forms will be used in the rest of this paper; for the present we note some overall qualitative features:
1. Mass has shifted to lower at the ILR (left panel), and to higher at the OLR (right panel), corresponding to at the ILR and at the OLR from Equation (51). Since the of every star is constant, the mass shift is due to the changes in the actions, of the resonant stars, with at the ILR and at the OLR; see footnote 2.
2. is larger than , because the Mestel DF is larger at the ILR than the OLR; see Figure 3. If this overall difference in magnitudes is factored out, we see that the isocontours of and are, very roughly, mirror-images of each other, when are expressed in units of the corresponding (i.e. and , respectively).
3. The scar at the CR (central panel) is very different. is an odd function of , and has smaller magnitude. is conserved during the mass shifts, because it is equal to the fast action at the CR. Moreover, for J_{R}<\mbox{L_{}}/\beta\simeq 0.03\mbox{L_{}}, and for J_{R}>\mbox{L_{}}/\beta\simeq 0.03\mbox{L_{}}, with from Equation (51). The changes in lead to the radial mixing of stars (Sellwood & Binney, 2002); gives a phase space picture of this process for a transient mode.
4.4.2 Surface densities of scars
The DF of each scar, , gives rise to an axisymmetric perturbation in the disk surface density, . This is obtained by integrating over velocity space:
[TABLE]
where Equation (37) may be used express to as a function of .
Since the DF of the scar is localized around L_{z}\simeq\mbox{L_{*}}, we expect the surface density of the scar, , to be localized around the resonant radius
[TABLE]
The standard epicyclic approximation (R_{\rm epi}\ll\mbox{R_{*}}) — which was sufficient to calculate from for the unperturbed disk — cannot be applied directly to evaluate the integral in Equation (58). This is because , the half-width of the scar in phase space, introduces a new small radial scale in the problem:
[TABLE]
Comparing with the typical epicyclic radius of orbits in the scar,
[TABLE]
where (\Delta J/\mbox{J_{}}) is the nonlinearity of the scar given in Equation (46). For a Mestel disk with , and an transient with \mbox{\varepsilon_{\rm d}}=10^{-2} — used in the estimates of (\Delta J/\mbox{J_{}}) in Equation (47) — we have
[TABLE]
at the ILR, CR and OLR, respectively.
With this estimate in hand, we see that the appropriate generalization of the epicyclic approximation is the ordering \Delta R\ll R_{\rm epi}\ll\mbox{R_{*}}, even though it is not so good at the ILR and CR. This is applied to the integral of Equation (58) in Appendix B.1 to obtain:
[TABLE]
where is a resonance-dependent dimensionless number, \Sigma_{0}(\mbox{R_{}})=\eta V_{0}^{2}/2{\rm\pi}G\mbox{R_{}} is the surface density of the unperturbed Mestel disk at , and is an dimensionless function that contains all the -dependence.
[TABLE]
determines the magnitude and sense of mass shifts due to the surface density perturbation. For the same parameter values used for the estimates in Equation (61),
[TABLE]
at the ILR, CR and OLR, respectively; it is negative at the ILR and positive at the CR and OLR. The shape function,
[TABLE]
is a function of (R/\mbox{R_{}}) because R_{\rm epi}\propto\mbox{R_{}}. The Gaussian factor implies that the radial scale of the scar is the local epicyclic radius.
The fractional surface density, \Delta\Sigma_{\ell m}(R)/\Sigma_{0}(\mbox{R_{*}})=\mu_{\ell m}\,S(R), is plotted at the principal resonances in Figure 5. This illustrates the following general properties of the surface densities of scars.
1. vanishes at , and shows opposite behaviour inside/outside ; more precisely, is an odd function of (R-\mbox{R_{}}). The total mass in the scar, , is given by the sum of two terms; one of them is and the other is . Since the integrands are odd functions of , both the integrals are vanishingly small in the epicyclic limit, \mbox{R_{}}\gg R_{\rm epi}\,. So the total mass is every scar is zero, as expected.
2. Since , the width of the scar in real space is due to epicyclic broadening, with the information about the “intrinsic” width, , now contained in the constant .
3. The sign of determines the sense of mass shifts across . It is radially inward when , and radially outward when . Hence mass shifts outward at the CR and, for for not too large, inward/outward at the ILR/OLR. The mass shifts are largest at corotation and cause radial mixing of stars (Sellwood & Binney, 2002), even though the angular momentum absorbed at the CR is, to leading order, zero from Equation (51).
5 SUPPRESSED TORQUES IN THE FINAL DISK
In § 3 we considered the nonlinear transformation of a smooth initial axisymmetric DF to a scarred final axisymmetric DF, due to the passage of an adiabatic, non-axisymmetric transient mode with angular wavenumber and pattern speed . The final DF is equal to the initial DF everywhere in phase space, except in the vicinity of resonances, as given in Equation (27). Both are axisymmetric DFs, but the initial DF is smooth, whereas the final DF is flattened near resonances; see Figure 2. In this section we demonstrate that this is responsible for the very different responses of the two DFs, to any new linear perturbation with the same angular wavenumber and pattern speed as the original transient mode.
In § 5.1 we formulate the problem of comparing the LBK torques, in the initial and final DFs, by casting the formulae of § 2.3 in terms of slow and fast action-angle variables. This reveals a very interesting property that is entirely a result of the phase space structure of the final DF: when the self-gravity due to the scars is neglected in the final axisymmetric disk, all the LBK torques, , vanish. Hence, the extra radial acceleration due to the gravity of the scars must be considered. This causes small changes in the radial and angular frequencies, and leading to slight shifts of resonances, which are calculated in § 5.2 for the scarred Mestel disk of § 4. In § 5.3 we derive a formula , which is non-zero but still much smaller than of Equation (70). The consequences of suppressed resonant torques for mode renewal are then addressed.
5.1 LBK torques in the initial and final DFs - I
Here we discuss the LBK torques acting on the initial and final DFs, due to a linear mode with potential perturbation,
[TABLE]
Since the mode is linear, \mbox{\varepsilon_{\rm s}}\ll 1 is infinitesimal, and for a perturbation that is applied gradually in time. and give the radial profile of the mode amplitude and phase, and can be of general form.
We consider first the initial disk with DF, , and surface density profile . The gravity of (and any external source such as a dark halo) gives rise to a radial acceleration profile, , which determines the radial and angular frequencies, and given in Equation (7). The torque due to a marginally growing, i.e. , at the resonance is given by Equation (13b):
[TABLE]
where \mbox{\widetilde{\Psi}}_{\ell m}(J_{R},L_{z}) are Fourier coefficients corresponding to . Since the contribution to the integral is only from the resonant curve in the plane, we can rewrite the integral in terms of the fast and slow action-angle variables, which are valid coordinates in the vicinity of the resonant line. From Equation (24), the initial DF is \mbox{F_{\rm in}}(\mbox{J_{\rm f}},\mbox{J_{\rm s}})=F_{0}(\mbox{J_{\rm f}}+\ell\mbox{J_{\rm s}}\,,\,m\mbox{J_{\rm s}}). Defining \mbox{\widehat{\Psi}}_{\ell m}(\mbox{J_{\rm f}},\mbox{J_{\rm s}})=\mbox{\widetilde{\Psi}}_{\ell m}(\mbox{J_{\rm f}}+\ell\mbox{J_{\rm s}}\,,\,m\mbox{J_{\rm s}}), we have
[TABLE]
We write \delta(\Omega_{\rm s})=\delta(\mbox{J_{\rm s}}-\mbox{J_{}})/|B_{*}(\mbox{J_{\rm f}})|, where \mbox{J_{\rm s}}=\mbox{J_{}}(\mbox{J_{\rm f}}) is the equation of the resonant curve of Equation (20), and B_{*}(\mbox{J_{\rm f}})=(\mbox{\partial}\Omega_{\rm s}/\mbox{\partial}\mbox{J_{\rm s}})_{J_{\rm s}=J_{*}} as given for the pendulum Hamiltonian of Equation (22). Integrating over , we obtain a compact expression for the resonant torque:
[TABLE]
where F_{\rm in}^{(n)}(\mbox{J_{\rm f}})=(\mbox{\partial}^{n}\mbox{F_{\rm in}}/\mbox{\partial}J_{\rm s}^{n})_{J_{\rm s}=J_{*}}. The evaluation of has been reduced to a one dimensional integral over , from its minimum value of of Equation (33) to . This formula is valid for a smooth initial DF of general form, and not restricted to the cool Mestel disk.
We want to derive a similar expression for the torque exerted by on the scarred disk, described by the final (post-transient) DF. The final DF is equal to the sum of the initial DF and the DFs of scars at all the resonances. So the surface density profile of the final disk is
[TABLE]
where is the surface density due to the scar at the resonance. The radial acceleration profile in the final disk is
[TABLE]
where is the radial acceleration due to the gravity of . The magnitude of these changes is of order the fractional perturbation, .
For the cool Mestel disk, is very small; see Equation (78). So it is reasonable to begin with the lowest-order estimate of resonant torques in the final disk, by setting . This is equivalent to neglecting , so . Then the action-angle variables for the final disk, , as well as and , are the same as for the initial disk. Hence the resonant curves, determined by \ell\Omega_{R}+m\!\left\{\Omega_{\phi}-\mbox{\Omega_{\rm p}}\right\}=0, are also the same for the two disks.
As earlier, we can introduce (\mbox{J_{\rm f}},\mbox{J_{\rm s}}) variables near the resonance. Then the resonant torque due to in the final disk is
[TABLE]
where F_{\rm fin}^{(1)}(\mbox{J_{\rm f}})=(\mbox{\partial}\mbox{F_{\rm fin}}/\mbox{\partial}\mbox{J_{\rm s}})_{J_{\rm s}=J_{*}}. We can use Equation (29) to express \mbox{F_{\rm fin}}(\mbox{J_{\rm f}},\mbox{J_{\rm s}}) in terms of \mbox{F_{\rm in}}(\mbox{J_{\rm f}},\mbox{J_{\rm s}}). Then F_{\rm fin}^{(1)}(\mbox{J_{\rm f}})=F_{\rm in}^{(2)}(\mbox{J_{\rm f}})(\mbox{J_{\rm s}}-\mbox{J_{*}})|_{J_{\rm s}=J_{*}}=0, which can also be seen from Figure 2, where the final, flattened DF has zero slope at resonance. Then , so the linear mode exerts no torque on disk stars. This conclusion holds for a scarred, final DF of general form, and not restricted to the scarred Mestel disk.
The vanishing of all resonant torques is due to a conjuction of two ingredients: (i) the limit , which ensured that the resonant lines in the final disk coincide exactly with those in the initial disk; and (ii) that the final DF has exactly zero slope at the scar center. When , the resonant lines in the final disk will be slightly shifted from those in the initial disk, where the slope of the final DF would be non-zero, resulting in non-zero .
5.2 Resonance shifts in the scarred Mestel disk
Since the theory gives a vanishing result for , we are obliged to calculate to the next order and consider a small but non-zero , such as those given in Equation (65). We want to calculate the orbital and epicyclic frequencies, and , inside the scars of the final disk, which are defined in terms of , the radial acceleration profile in the scarred disk. We must first determine the new guiding-centre radius, , by solving . Then
[TABLE]
gives the frequencies in the scarred Mestel disk. From Equation (72), is the sum of the known function and at all the resonances, which we now calculate.
5.2.1 within a scar
From Figure 5, we see that is concentrated within an annulus of half-width about the resonant radius, outside which its amplitude falls rapidly. So, at distances much larger than its width, \left|R-\mbox{R_{*}}\right|\gg R_{\rm epi}, the scar appears as two concentric circular wires near with equal and opposite masses. is due mainly to the (dipolar) far-field, and is small because of close cancellations between the attraction and repulsion of the two wires. Since the resonant radii are well-separated for different , the field of one scar on another can be neglected in a first approximation. Hence the dominant contribution to near a scar is due to its self-gravity.999A scar can be thought of as a ring-like “gravitational capacitor” in real space, with its external field smaller than its internal field. Equation (75) relates the internal field to the surface density profile of the scar. We have
[TABLE]
where is given in Equation (63). This formula is exactly equivalent to the familiar relationship between the potential and surface density of tightly-wound spiral patterns — see Equations (6.29) and (6.30) of Binney & Tremaine (2008).
This integral can be expressed in terms of special functions, but we do not need these for the purposes of calculating frequency shifts close to resonance. From Equation (74) we see that depends on , whereas depends on both and . Hence we require only to first order in (R-\mbox{R_{}})/\mbox{R_{}}\,. We calculate this in Appendix B.2, to obtain:
[TABLE]
where
[TABLE]
is a dimensionless resonance-dependent number that is a measure of the fractional difference in the radial acceleration near , between the final and initial Mestel disks. For the same parameter values used for the estimates of in Equation (65), we have
[TABLE]
at the ILR, CR and OLR, respectively.
5.2.2 Frequency profiles within a scar
Having determined , we can calculate the guiding-centre radius, , by solving . We require this only within the scar in phase space, i.e. for |L_{z}-\mbox{L_{*}}|<\Delta L. Solving to first order in , we get . The fractional shift in the guiding centre radius, , is negative at the ILR and positive at the CR and the OLR. The frequencies can now be calculated by using in Equation (74):
[TABLE]
We can get the frequencies to first order in by setting in the second terms on the right side. Using Equation (76) for , we obtain
[TABLE]
The slow frequency within the scar is
[TABLE]
Similar to the slow frequency in the initial Mestel, \,\Omega_{\rm s}(\mbox{J_{\rm s}}) of Equation (43), \Omega^{\prime}_{\rm s}(\mbox{J_{\rm s}}) is a function of (\mbox{J_{\rm s}}/\mbox{J_{*}}). But its functional form is resonance-dependent, as can be seen in Figure 6. Indeed the fractional difference between and is equal to , which is positive at the ILR and negative at the CR and the OLR.
The equation of the resonant line in the final disk is given by \mbox{J_{\rm s}}=J^{\prime}_{*}, where . Solving this to first order in , we obtain
[TABLE]
which shows that resonant lines undergo shifts \delta J_{*}=-(\delta_{\ell m}/2)\mbox{J_{*}}. At the shifted resonant location, the quantity
[TABLE]
For the same values of parameters as earlier, the fractional shifts at the principal resonances are
[TABLE]
This is positive at the ILR and negative at the CR and the OLR; see Figure 6. The scar half-widths, of Equation (47), are two orders of magnitude larger than the shifts. Since , the shifted resonant lines lie well within the scars where the final DF is flattened. Below we show the non-zero shifts of ILR/CR/OLR give rise to non-zero resonant torques in the final disk; since the shifts are small, the torques are weaker than in the initial disk.
5.3 LBK torques in the initial and final DFs - II
We are now in a position to compare resonant torques in the initial and final disks, due to the linear mode of Equation (67), taking into account the self-gravity of the scars. In order to make quantitative estimates, we use the smooth and scarred Mestel disks.
5.3.1 Torques in the smooth Mestel disk
From Equation (70), the resonant torque in the smooth Mestel disk is:
[TABLE]
where |B_{*}|=m\mbox{\Omega_{\rm p}}/\mbox{J_{}}, and F_{\rm in}^{(1)}(\mbox{J_{\rm f}}) is given in Equation (49). It is convenient to use the integration variable x=(\beta/m)\left(\mbox{J_{\rm f}}/\mbox{J_{}}+\ell\right), and Fourier coefficient \psi_{\ell m}(x,\beta)=\mbox{\widehat{\Psi}}_{\ell m}(\mbox{J_{}}(mx/\beta-\ell),\,\mbox{J_{}}). Then
[TABLE]
where
[TABLE]
is the torque weight function for the smooth Mestel disk, plotted in the left panel of Figure 7, at the principal resonances.
5.3.2 Torques in the scarred Mestel disk
We will derive an analogous equation for the resonant torques due to on the scarred Mestel disk, taking into account the shifts in frequencies and resonance locations due to the self-gravity of the scars, derived in § 5.2. Let be the action variables in the scarred disk. , but of Equation (37) within the scar. Here, it is given by
[TABLE]
where is given in Equation (80) and . The slow and fast actions near the resonance are \mbox{J^{\prime}{\rm f}}=J^{\prime}_{R}-(\ell/m)L^{\prime}_{z} and \mbox{J^{\prime}{\rm s}}=L^{\prime}_{z}/m. Since and differ from and by , Equation (88) gives . Then \mbox{J^{\prime}{\rm f}}=\mbox{J{\rm f}}+O(\delta_{\ell m}), whereas \mbox{J^{\prime}{\rm s}}=\mbox{J{\rm s}} remains unchanged. The shift in the fast action is unimportant, and we can set (\mbox{J^{\prime}{\rm f}},\mbox{J^{\prime}{\rm s}})\to(\mbox{J_{\rm f}},\mbox{J_{\rm s}}).
The resonant torque is
[TABLE]
where \,\Omega^{\prime}_{\rm s}(\mbox{J_{\rm s}}) is the slow frequency of Equation (81), and \mbox{F_{\rm fin}}(\mbox{J_{\rm f}},\mbox{J_{\rm s}}) is given in terms of the initial DF by Equation (29). We want to calculate the integrand to leading order in :
[TABLE]
where we have used Equation (83) for . From Equations (47) and (84) we know that the shifted position of the resonance lies well inside the scar in phase space. Using this in Equation (29), we have \,\mbox{F_{\rm fin}}(\mbox{J_{\rm f}},\mbox{J_{\rm s}})\simeq F^{(0)}_{\rm in}+(1/2)F^{(2)}_{\rm in}(\mbox{J_{\rm s}}-\mbox{J_{*}})^{2}. Hence, the integrand is
[TABLE]
From Equation (82), we have \mbox{J^{\prime}{*}}-\mbox{J{}}\simeq-(\delta_{\ell m}/2)\mbox{J_{}}. Also, \,\mbox{\widehat{\Psi}}_{\ell m}(\mbox{J_{\rm f}},\mbox{J^{\prime}{*}})=\mbox{\widehat{\Psi}}_{\ell m}(\mbox{J{\rm f}},\mbox{J_{*}})+O(\delta_{\ell m}). Therefore, to first order in , the integrand is given by
[TABLE]
Substituting Equation (90) in (89), and integrating over , the torque in the scarred Mestel disk is:
[TABLE]
This is similar in form to of Equation (85), the torque in the smooth Mestel disk. The difference is that the integral has acquired a pre-factor , and \mbox{J_{*}}F_{\rm in}^{(2)} has replaced in the integrand. Since , we can guess that . But this needs to be established by calculating F^{(2)}_{\rm in}(\mbox{J_{\rm f}})=(\mbox{\partial}^{2}F_{\rm in}/\mbox{\partial}J^{2}_{\rm s})_{J_{\rm s}=J_{*}} for the Mestel DF of Equation (48):
[TABLE]
Introducing the integration variable , used in the passage from Equation (85) to (86), we obtain
[TABLE]
where
[TABLE]
is the torque weight function for the scarred Mestel disk, plotted in right panel of Figure 7, at the principal resonances.
5.3.3 Torque suppression factor
We want to make quantitative estimates of how small is compared to ; to this end we define the torque suppression factor due to a scar, . Equation (92) for is similar in form to Equation (86) for . Both torques have the same function, , in the integrand, which depends on the radial profile of the linear mode, . As discussed earlier, this must be taken from either models of observational data or fits to numerical simulations. However, we do not need to know the detailed functional form of in order to make a rough estimate of . This is done below by exploiting some specific properties of the weight functions, of Equation (87) and of Equation (93), for the smooth and scarred disks, respectively.
Both weight functions fall off exponentially with , so smaller values of contribute most to the integrals. As can be seen in Figures 7 and 8, and are negative at the ILR and positive at the OLR, so the linear mode absorbs/emits angular momentum at the ILR/OLR, as expected. Moreover, both weight functions have qualitatively similar forms as functions of , but is smaller than by a factor of about for the ILR and for the OLR. Both weight functions are small at the CR, as expected, and is smaller than by a factor of about . Therefore, for of general form, will be much smaller than . Hence, a first estimate of the ratio of the torques is given by the ratio of their weight functions near . Therefore, the torque suppression factor is:
[TABLE]
Using the values for in Equation (78), and and as earlier,
[TABLE]
at the ILR, CR and OLR, respectively.
6 RENEWAL OF MODES IN THE SCARRED DISK
In § 6.1 we first establish that the suppressed torques, exerted on the final disk by linear modes of the form of Equation (67), also imply that the heating of disk stars is much smaller in the final disk than in the initial disk. In contrast, for some other linear mode , with a different angular wavenumber or pattern speed, resonant torques and heating are generically comparable in the initial and final disks. In § 6.2 we consider the implications of this for mode renewal in the final disk, and compare with the simulations of SC14.
6.1 Suppressed epicyclic heating due to
The epicyclic energies (per unit mass) of resonant stars change at the rates and in the initial and final disks, respectively. Here and are the epicyclic frequencies at resonance. Both and are positive at the ILR and OLR, and vanish at the CR. Since , we expect that . This is indeed true: since and differ only by , to first order in , the ratio of the heating rates is
[TABLE]
Therefore, linear modes of the form, of Equation (67), suffer much less collisionless dissipation in the final disk. This is entirely due to the fact that resonance shifts are small, , so that the resonant lines lie within the scars of the final disk, where the final DF is flattened (as a function of ).
We now compare the torques in the initial and final disks, due to some other linear mode, :
[TABLE]
where and can be different from and . Below we show that (i) the two sets of ILR/CR/OLR, for and , are well-separated in the smooth and scarred Mestel disks; (ii) Generically, the ILR/CR/OLR of pass through scar-free regions of the final disk, so that resonant torques and heating due to will be nearly the same in both disks.
1. We first consider resonant lines, due to and due to in the smooth Mestel disk. The latter is given as \mbox{J_{\rm s}}=\mbox{J_{*}} of Equation (42), where we will treat the parameters (m,\mbox{\Omega_{\rm p}}) as given and fixed. Similarly, the resonant line is \mbox{J_{\rm s}}=\bar{J}_{*}, where
[TABLE]
In order for the resonant lines of and to be close to each other, \bar{J}_{*}\simeq\mbox{J_{*}}. For the CRs (i.e. ) this implies that \bar{m}\bar{\Omega}_{\rm p}\simeq m\mbox{\Omega_{\rm p}}, which condition can evidently be satisfied by many and \bar{\Omega}_{\rm p}\neq\mbox{\Omega_{\rm p}}. In addition, if we require that the ILRs and OLRs also must coincide, . Since and are positive integers this can be satisfied only if and \bar{\Omega}_{\rm p}\simeq\mbox{\Omega_{\rm p}}. Hence, for or significantly different from , the and resonant lines will be well-separated in phase space.
2. Next, we consider resonant lines, due to and due to in the scarred Mestel disk. The latter is given as \mbox{J_{\rm s}}=J^{\prime}_{*}=\left[1-(\delta_{\ell m}/2)\right]\!\mbox{J_{*}}, which we know lies well within the scar. Outside the scarred regions, the orbital and epicyclic frequencies are nearly the same in both disks (with small differences due to the dipolar fields of distant scars, discussed in § 5.2.1). So the resonant line in the scarred disk will nearly coincide with the line in the smooth disk.
3. From items 1 and 2 above, we conclude that the ILR/CR/OLR due to lie close to each other in the smooth and scarred disks, and that these must be well-separated from the ILR/CR/OLR due to . Hence, the line in the final disk generically traverses scar-free regions of phase space,101010We note the possibility that a particular resonant curve of might pass through some scar, while other resonant curves of (with ) lie in scar-free regions. We assume that this is not the case, but that does not mean that such an “accidental overlap” of an single resonant line with a single scar is uninteresting. Indeed, it might well prove important, but this problem is beyond the scope of this paper. where the DF near it is given by the smooth Mestel DF of Equation (38), the resonant torques exerted by a marginally growing are nearly the same for the smooth and scarred Mestel disks, and given by
[TABLE]
where is the same as in Equation (87) with replaced by , and is the Fourier coefficient corresponding to . Equivalently, we can think of the torque suppression factor, . From Equation (96), the ratio of the heating rates in the final and initial disks is , so can be expected to cause comparable amounts of heating in both disks.
6.2 Model of mode renewal
In numerical simulations of smooth stellar disks, sampling shot-noise generates a spectrum of spiral modes, some of which grow to somewhat large amplitudes and decay through absorption at the Lindblad resonances. Let us consider one of these transient spiral mode, with angular wavenumber and pattern speed . It will transform an initially smooth disk into a scarred disk, whose main scars are at the ILR/CR/OLR. Spiral modes generated by shot-noise in the scarred disk will behave differently. In § 6.1 we showed that the resonant torque and heating suppression factor for linear modes, , with the the same (m,\mbox{\Omega_{\rm p}}) as the precursor transient mode. But for a linear mode with different , for all , except in the case of accidental overlaps mentioned in footnote 10. Hence, among the spectrum of modes generated by shot-noise, modes like — that suffer far less dissipation than other linear like — will grow to larger amplitudes and dominate the renewed spiral pattern. We may think of the scars as filtering a noisy generator.
There appears to be some support for this model of mode renewal in the simulations of SC14. Their Figure 5 plots the power in modes as functions of radius and frequency. The top panels correspond to transient modes that scar the disk, and the bottom panels are for modes that are regenerated in the scarred disk. The central panels are for the mode, which show that the renewed mode has the same frequency as its precursor. Moreover, their spatial forms, displayed in panels (c) and (f) of Figure 6, are also strikingly similar. But the interpretation is not so clean, because of the presence of large-amplitude modes; this is briefly discussed in § 7.
SC14’s physical picture of mode renewal is presented in the “local” picture of wavepacket behaviour near Lindblad resonances:
“The scattering of stars as each wave decays takes place over narrow ranges of angular momentum, causing abrupt changes to the impedance of the disk to subsequent traveling waves. Partial reflections of waves at these newly created features allows new standing-wave instabilities to appear that saturate and decay in their turn, scattering particles at new locations, creating a recurring cycle.”
In the “global” LBK approach of our paper, the “narrow ranges of angular momentum” would correspond to the narrow scars surrounding resonant curves in phase space; the “abrupt changes to the impedance of the disk to subsequent traveling waves” can be thought of as the abrupt flattening of the final DF (as a function of the slow action) across a scar; and the “partial reflection of waves…” would be due to the suppressed dissipation within scars. The renewed modes in our model would then correspond to the cavity-type “mirror modes” described by SC14, which are sustained by swing amplification at the CR and partial reflection at the scar of a Lindblad resonance.
The partial reflection must, as SC14 realize, arise from a modification near the Lindblad resonances, of the WKB dispersion relation for spiral density waves. We can think of this in the following manner. The standard Lin-Shu-Kalnajs dispersion relation for stellar disks (Lin & Shu, 1966; Binney & Tremaine, 2008) is valid for the smooth initial DF. In order to study reflection of waves in the scarred disk, it is necessary to derive a dispersion relation for spiral wavepackets in the final disk. Since our final DF is equal to the initial DF outside scars and flattened within, we may expect the new dispersion relation to be different from the old one only near resonances. This can be worked out; indeed, it is necessary to supplement our “global” LBK approach with a “local” picture of wavepacket propagation between the corotation and Lindblad resonances. Only then will we understand weak torques, suppressed dissipation and mode renewal in terms of spiral density waves that are partially reflected at the Lindblad resonances. What may we expect of such a modified LSK dispersion, based on the results of § 5 and 6.1? In § 5.1 we showed that, when the self-gravity of the scars is neglected (i.e. when ), the resonant torques vanish and there is no dissipation. In the corresponding local picture, the encounter of a wavepacket with a resonant scar must be lossless, so reflection should be perfect. When self-gravity is included (i.e. when ) the resonant torques and dissipation are suppressed but not zero, as we showed in § 6; this more generic case should correspond to partial reflection of a wavepacket.
SC14 also observe that each cycle of mode renewal heats up the disk until , when it becomes less responsive. This fact is also consistent with our picture of suppressed dissipation of certain linear modes of the form . This is because, in our model of mode renewal as a noise-filtering process, the lower dissipation rates of these modes enable them to grow to greater amplitudes than other linear modes . But the decay of these, by now, large-amplitude modes would be governed by nonlinear effects; SC14’s simulations appear to suggest that the nonlinear decay time scales may be comparable in the initial and final disks, but this remains to be investigated. We suggest that the filtering action of scars, operating on noise-generated modes when their amplitudes are still small enough to be “linear”, is the basic physical mechanism regulating mode renewal.
7 CONCLUSIONS
We have presented a model of the renewal of non-axisymmetric modes in stellar disks that have experienced the passage of a small-amplitude, transient non-axisymmetric mode with angular wavenumber and pattern speed . The physical mechanism relies on the nonlinear readjustment of the phase space distribution of stars in the vicinity of resonances, which can be thought of as scars left behind by the transient mode. The DF of the final disk is flattened (as a function of the slow action) within narrow scars, resulting in the suppression of resonant torques and epicyclic heating. In particular, linear modes with the same (m,\mbox{\Omega_{\rm p}}) as the transient mode that produced the final disk from the initial disk transport much less angular momentum, and suffer much less dissipation, than other linear modes with different angular wavenumbers or pattern speeds. Therefore, the set of resonant scars acts as a filter of the spectrum of linear modes generated by shot noise in numerical simulations, promoting the preferential growth of linear modes with the same (m,\mbox{\Omega_{\rm p}}). These then grow to large amplitudes and dominate the appearance of the disk. Their subsequent decay would depend on nonlinear processes that are not considered in this paper, and decay time scales could be comparable in the initial and final disks. Since the disk eventually heats up and becomes less responsive, some form of cooling is necessary for recurrent spiral activity; accretion of cold gas and new star formation have been thought of as the main physical agents, and we have little to add to this.
The renewed modes we identified correspond to the cavity-type mirror modes of SC14. In the local picture of wavepacket propagation, these can be thought of as spiral density waves that are swing-amplified at corotation and partially reflected at a Lindblad resonance in a scarred disk. These two physical ingredients have been emphasised recently by Binney (2019): “The swing amplifier and resonant absorption are the stand-out pieces of physics in this beautiful mechanism by which galaxies like ours evolve.”. Whereas the swing amplifier is reasonably well-understood, resonant absorption in scars has not received much attention. Our study of the physics of resonant absorption from a global LBK point of view has revealed that it can be highly suppressed within scars. We also noted that this should imply partial reflection of wavepackets in the local description, whose quantitative description should come from a modified LSK dispersion relation near scars, giving rise to abrupt changes in the impedance of the disk proposed by SC14.
The final DF for the scarred Mestel disk of § 4.4.1 could be used for initial conditions in simulations, to test our model of mode renewal. This will help clarify matters in at least two ways:
a. Our discussion of noise-generated modes in a scarred disk was based on simulations. The physical kinetics of the generation of phase structures, such as ridges in action space, from the discreteness of unscarred disks has been explored by Fouvry et al. (2015a, b). To study the filtering action of scars we have proposed, it is necessary to extend the theory to scarred disks. The theory could then be compared with controlled simulations of the scarred Mestel disk.
b. As support for our model, we cited the case of the renewal, recurrence even, of an mode in the simulations of SC14, while noting that the interpretation is confused by the presence of modes. SC14 state that these two modes are not independent, so it is plausible that there is some overlap in the sets of scars produced by the precursor modes, and nonlinear interactions among the modes. In the SC14 simulations, what we have referred to as the scarred disk was produced from the smooth disk through the action of a superposition of transient spiral and bar-like modes, of which the are the most prominent. Using the scarred Mestel disk would provide a cleaner initial condition.
Sellwood & Carlberg (2019) present numerical experiments on mode recurrence in disks seeded by a groove scored by hand. The groove gives rise to an instability, which is not a cavity-type mode (Sellwood & Lin, 1989; Sellwood & Kahn, 1991). This “groove mode” is transient, and its decay leaves behind scars at the principal resonances, as given in their Figure 6. The sense of mass shifts across resonant lines appears to be consistent with what is expected from our § 4.4.1; Figure 7 of their paper is essentially the DF of the OLR scar (when due allowance is made for the finite slopes of the resonant lines — see our footnote 7 of § 4.1). Two new modes are then generated by the scars of the transient precursor groove mode. The weaker of the two renewed modes has the same frequency as the precursor (lines 4 and 6 of their Table 2), and is likely to be a cavity-type mirror mode of the sort we discussed. But the stronger mode has a lower frequency than the precursor groove mode (lines 2, 3 and 5 of their Table 2), so it is not a cavity mode of the type we considered. However, its CR passes through the OLR scar of the precursor, implying that its Lindblad resonances would pass through unscarred parts of the disk. This may correspond to a case of “accidental overlap”, mentioned in footnote 10 of our § 6.1 but not discussed further.
We considered a precursor transient mode with fractional surface density perturbation, \mbox{\varepsilon_{\rm d}}=10^{-2}, but this can be increased to \mbox{\varepsilon_{\rm d}}=10^{-1}, which is closer to simulations, as argued below.111111The reason for not using the more realistic value, \mbox{\varepsilon_{\rm d}}=10^{-1}, straightaway is the following. Unless the pendulum approximation of Equation (22) itself needs improvement, we expect all the calculations upto and including § 4.4.1 would remain unchanged. But the formula for the surface density of scars, of Equation (63), would be modified. This was derived in Appendix B.1 by assuming that . From Equation (61), this is for \mbox{\varepsilon_{\rm d}}=10^{-2}. Since , this ratio for \mbox{\varepsilon_{\rm d}}=10^{-1}. Then , the radial acceleration due to scars, , and subsequent development, would require the use of special functions and integrals over them. It was felt that presenting these as such would have obscured the physics. Since the filtering action of scars arises from the flattening of the DF within scars, the resonance shifts, \delta\mbox{J_{}} of Equation (84), must be smaller in magnitude than , the half-width of the scar. For \mbox{\varepsilon_{\rm d}}=10^{-2}, we have |\delta\mbox{J_{}}|/\Delta J\sim 10^{-2} (for an mode), so resonant lines in the scarred disk pass through the very central regions of the scar. Since \delta\mbox{J_{}}\propto\varepsilon^{3/2}_{\rm d} and , we expect |\delta\mbox{J_{}}|/\Delta J\sim\mbox{\varepsilon_{\rm d}}. Hence, for \mbox{\varepsilon_{\rm d}}=10^{-1}, which is closer to simulations, |\delta\mbox{J_{*}}|/\Delta J\sim\mbox{few}\times 10^{-1}. The resonant lines still sample the flattened regions within scars, so Equation (94) for , the torque (and dissipation) suppression factor, would be valid. From Equation (95), at the principal resonances for \mbox{\varepsilon_{\rm d}}=10^{-2}. Since , for \mbox{\varepsilon_{\rm d}}=10^{-1}, we have . Resonant torques and dissipation are larger, but the filtering action of scars might still be expected to work for this more realistic transient mode.
There is room for further improvement in the calculations presented here, for closer comparisons with simulations or confrontation with observational data. We assumed that the growth and decay times of the transient mode were large enough that we could calculate the form of a scar in the adiabatic limit. But the transient modes in simulations are faster, so we need to extend our calculations beyond the adiabatic limit. The passage of a non-adiabatic transient will result in greater mixing of the initially axisymmetric DF. This will produce a scarred disk that is non-axisymmetric, unlike the adiabatically scarred disk we have considered in this paper. The non-axisymmetry is of interest, in itself, but it does not seem to affect mode renewal, as has been tested in simulations by starting re-runs after randomizing in at fixed , and (see e.g. Sellwood & Lin, 1989; Sellwood, 2012). So, we can focus on the axisymmetric part of the scars, and ask how different these are from the adiabatic scars of this paper; the more vigorous stirring would smoothen the boundaries, but the interesting questions would relate to DF changes near scar centers.
For explicit calculations involving scars in a cool Mestel disk, we replaced the functions D_{\ell m}(\mbox{J_{\rm f}}), determining the scar boundaries, by resonance-dependent constants . This proved a useful surrogate for the estimation of integrated quantities, providing results that are physically in line with expectations; the sense of mass shifts in phase space, the magnitudes and signs of the angular momentum absorbed during scar formation, and the shapes of which determine all the subsequent development. But it is a poor representation of the scar boundaries, and not suitable for comparison with observational data. Indeed, the functions D_{\ell m}(\mbox{J_{\rm f}}) can be calculated for any assumed radial profile for the transient. It is necessary to do this, were one to search for transient spiral activity in our Galaxy in GaiaDR2, such as Trick et al. (2019) for axisymmetric features and Sellwood et al. (2019); Monari et al. (2019) for non-axisymmetric features.
Coherent and symmetric spiral patterns are often seen in barred galaxies, or galaxies that have been tidally perturbed by interaction with a passing companion galaxy (Kormendy & Norman, 1979; Kendall, Kennicutt & Clarke, 2011). Since the spiral has a lower pattern speed than the bar (Sellwood & Sparke, 1989), this is a more complicated — and very interesting — problem than we considered, involving two pattern speeds. The tidal perturbation poses a different sort of problem, because the tidal potential does not have a well-defined pattern speed. But it drives a large-scale spiral pattern in the galaxy with a strong component that would eventually decay. If the tidally-driven spiral behaves somewhat like the transient modes of this paper, relic scars left behind by its passage could enable the renewal of patterns resembling “grand design” spirals. Further extension of our model, guided by simulations, is required to address these problems.
I thank Jerry Sellwood for very useful comments, and Karamveer Kaur for a careful reading of the manuscript and contributing to Appendix B.2.
Appendix A Final DF after the passage of the adiabatic transient
We use ST96 to follow the time evolution of the initial DF, \mbox{F_{\rm in}}(\mbox{J_{\rm f}},\mbox{J_{\rm s}}) of Equation (24), due to an adiabatic transient mode whose time profile function is non-zero only between and : increases monotonically from zero at , reaches a maximum value of unity at , and then decreases monotonically to zero at . The -evolving DF is followed as a function of the fast action and the “adiabatic invariant” of Equation (25), and the final DF of Equation (27) is derived.
ST96 is valid for Hamiltonians of more general form than , for which can be asymmetric and expand or contract at different rates. As given in Table 1 of ST96, there are altogether six cases of separatrix crossings to consider. But, for the of Equation (22), are always reflection-symmetric about \mbox{J_{\rm s}}=\mbox{J_{*}}, so the number of cases reduces to two; see items 2 and 5 below.
It is convenient to use separate notation for trapped orbits (in region II), and circulating orbits (in regions I and III). In II, for the elliptic fixed point orbit. For the limiting librating orbit, just inside , we can calculate using Equation (23):
[TABLE]
is discontinuous across , because the pendulum phase space is divided. We use to describe the trapped orbits of region II, and to describe the circulating orbits of regions I and III. The time-evolving DF is written as in II and in I and III:
[TABLE]
As grows and decays, the range of values taken by the variables, and , varies with . Within these shifting boundaries, the functional forms of the DFs, and , also change with . Below we follow the time evolution of and through seven simple steps, and determine the final DF.
1. Initial state: at , so both collapse to the line \mbox{J_{\rm s}}=\mbox{L_{}} and . The are no trapped orbits, so we can ignore both and . For the circulating orbits we know from § 3.2 that K_{b}=\mbox{J_{\rm s}}-\mbox{J_{}}. Then
[TABLE]
2. Expansion: As increases, circulating orbits encountering the expanding are trapped into librating orbits. Let us pick a time between and .
Since orbits in regions I and III (defined at ) have not experienced separatrix crossing, is an adiabatic invariant, so retains the same form as Equation (A3):
[TABLE]
where K_{1}={\cal K}(\tau_{1};\mbox{J_{\rm f}}).
All the trapped orbits with have experienced separatrix crossing. We need for the limiting librating orbit, , just inside . Since expand symmetrically, according to case c of Table 1 in ST96, contains equal proportions of at :
[TABLE]
where we have used Equation (A4) to write the last equality.
3. Further expansion: As increases, expand and trap orbits with . Meanwhile, the value of the DF at remains frozen:
[TABLE]
4. Maximal expansion: Region II is maximal at when . The orbits in I and III (defined at ) do not ever experience separatrix crossing. For these, at the final time is the same as at the initial time. Using Equation (A3), we have
[TABLE]
where
[TABLE]
Below we calculate F_{b}(K_{b},\tau_{>};\mbox{J_{\rm f}}) for |K_{b}|<\Delta J(\mbox{J_{\rm f}}).
5. Contraction: For , is a decreasing function of , and contract. Trapped orbits encountering the contracting are liberated into circulating orbits. Let be the (unique) time between and , when . At this time, , and , just like at in item 2 above. According to case f of Table 1 in ST96, at (in I and III, respectively) is equal to F_{a}(K_{a}\to K_{1},\tau_{2};\mbox{J_{\rm f}}), which is the DF of the limiting trapped orbit at this time. Since the trapped orbit with has not experienced separatrix crossing during the interval , its DF has remained frozen in the form given by Equation (A6). Then
[TABLE]
6. Further contraction: As decreases further, shrink and the circulating orbits at (in I and III, respectivelt) do not experience any more separatrix crossing. Hence, for all , \,F_{b}(\pm K_{1}/2,\tau;\mbox{J_{\rm f}})=F_{b}(\pm K_{1}/2,\tau_{2};\mbox{J_{\rm f}}) remains frozen at the value given by Equation (A9). We now note that, in item 2, we could have chosen to take any value between and . So can take any value between [math] and . Hence
[TABLE]
7. Final state: Since , just like the initial state, there are no trapped orbits. The circulating orbits are described by the of Equations (A7) and (A10). Similar to the initial state, we have K_{b}=\mbox{J_{\rm s}}-\mbox{J_{*}} in the final state. Substituting this in Equations (A7) and (A10), we obtain Equation (27) for the final DF.
Appendix B Physical quantities for a scar in a cool Mestel disk
B.1 Surface density
Here we derive Equation (63) for , by using Equation (56) for in Equation (58). This is done in the small scar-width limit \Delta R\ll R_{\rm epi}\ll\mbox{R_{}}, which is equivalent to \Delta L\ll V_{0}R_{\rm epi}\ll\mbox{L_{}}. Then for \,|L_{z}-\mbox{L_{}}|\geq\Delta L\,, and for \,|L_{z}-\mbox{L_{}}|<\Delta L\,,
[TABLE]
In the small scar-width limit,
[TABLE]
where \kappa_{*}=\sqrt{2}V_{0}^{2}/\mbox{L_{*}} is the epicyclic frequency at resonance. We can use this in Equation (B1) to express as a function of , which is the form required to calculate
[TABLE]
The integral over involves Gaussian-type integrals, and is readily calculated. Then
[TABLE]
This integral can be evaluated in terms of special functions, but we do not need them in the small scar-width limit. Since the integrand is a smooth function, it can be expanded in a Taylor series about L_{z}=\mbox{L_{}}, where it can be seen that the lowest-order contribution comes from the (L_{z}-\mbox{L_{}})^{2} term. Integrating this gives
[TABLE]
The surface density of a scar is . is concentrated about and becomes small for |R-\mbox{R_{}}|\gg R_{\rm epi}, because of the Gaussian factor. Using the definitions of the constants , , R_{\rm epi}\simeq 2^{-1/4}\beta^{-1/2}\,\mbox{R_{}}, and Equation (57) for , we obtain as given in Equation (63).
B.2 Radial acceleration
Here we derive Equation (76) by substituting of Equation (63) in Equation (75):
[TABLE]
Changing to a dimensional integration variable, x=(R^{\prime}-\mbox{R_{}})/R_{\rm epi}\, and using x_{0}=(R-\mbox{R_{}})/R_{\rm epi}\,,
[TABLE]
where
[TABLE]
Since the Gaussian decays rapidly for and , the lower limits of integration can be taken as and \left[\,1+(R_{\rm epi}/\mbox{R_{}})\,x\,\right]^{-1}\simeq\left[\,1-(R_{\rm epi}/\mbox{R_{}})\,x\,\right]. Then
[TABLE]
can be written in terms of special functions. But we require and only to first order in :
[TABLE]
Substituting this in the above formula for gives Equation (76).
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