Constraints on Radial Migration in Spiral Galaxies - II. Angular momentum distribution and preferential migration
Kathryne J. Daniel Rosemary F. G. Wyse
astro-ph.GA



firstpage–lastpage 2018

Dust in brown dwarfs and extra-solar planets
    Mahesh Babu Vaddi and B. Sundar Rajan
 
 Department of Electrical Communication Engineering, Indian Institute of Science, Bengaluru 560012, KA, India 
 E-mail: {vaddi, bsrajan}@iisc.ac.in 
    Accepted January 20, 2018. Received January 3, 2018; in original form October 30, 2017
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The orbital angular momentum of individual stars in galactic discs can be permanently changed through torques from transient spiral patterns. Interactions at the corotation resonance dominate these changes and have the further property of conserving orbital circularity. We derived in an earlier paper an analytic criterion that an unperturbed stellar orbit must satisfy in order for such an interaction to occur i.e. for it to be in a trapped orbit around corotation. We here use this criterion in an investigation of how the efficiency of induced radial migration for a population of disc stars varies with the angular momentum distribution of that population. We frame our results in terms of the velocity dispersion of the population, this being an easier observable than is the angular momentum distribution. Specifically, we investigate how the fraction of stars in trapped orbits at corotation varies with the velocity dispersion of the population, for a system with an assumed flat rotation curve. Our analytic results agree with the finding from simulations that radial migration is less effective in populations with hotter' kinematics. We further quantify the dependence of this trapped fraction on the strength of the spiral pattern, finding a higher trapped fraction for higher amplitude perturbations.
 


galaxies: evolution, galaxies: kinematics and dynamics, galaxies: structure




§ INTRODUCTION


Internal, secular mechanisms can have a significant impact on the
evolution of galactic discs <cit.>.
For example, it is now recognized that disc stars can migrate
radially, moving of order a disc scale-length over their lifetime,
through successive resonant interactions with transient gravitational
perturbations (for observational evidence, see
e.g., <cit.>
 and for theoretical investigations see e.g. <cit.>). <cit.> established that stars
on close to circular orbits can exchange orbital angular momentum with
steady (or slowly growing) spiral waves at the major resonances,
namely corotation and the inner/outer Lindblad resonances.
These authors further demonstrated that the change in angular momentum
at corotation occurs without any increase in the energy
associated with non-circular motion, whereas torques at the Lindblad
resonances do increase orbital eccentricity (i.e. cause kinematic heating of the population).

<cit.>, in an influential paper, showed that for a star in a trapped orbit"[We use the term trapped orbit" for
members of the horseshoe" orbital family associated with the
corotation resonance, in order to emphasise their role as the first
stage in radial migration.] around the corotation radius of a
transient spiral pattern, the angular momentum change can be permanent, leading to radial migration of the star.  While the spiral pattern exists, a trapped star is
repeatedly torqued, causing its 
mean orbital radius (hereafter denoted R_L, under the assumption that it  equals the guiding
centre radius in the epicyclic approximation, i.e. the radius of a
circular orbit with angular momentum equal to that of the star at that instant) to
move outwards then inwards across corotation. The star's guiding centre radius oscillates in time until the torques cease when the
transient spiral perturbation dies away, restoring axisymmetry and leaving the star with a new value for  orbital angular momentum and guiding radius (except in the unlikely case that
it is dropped back where it started). There is  no associated increase in
non-circular motion. <cit.> showed that interactions at the
corotation resonance dominate the angular momentum changes
in the disc (compared to interactions at the Lindblad
resonances). We will refer to the changes in mean orbital radius
due to the corotation interactions as radial migration”.

All orbits have an associated circular frequency and so, provided
there exists an appropriately matched spiral (with pattern speed approximately equal to the stars's circular frequency) for a long enough period
of time, any star could in principle migrate radially.  However, as
pointed out by <cit.>, stars on eccentric orbits cannot keep station" with the spiral pattern at the radius of corotation and
therefore radial migration is expected to be less important for
kinematically hotter stellar populations.  Our investigations in this
paper are aimed at quantifying this expectation.

We earlier <cit.> derived an analytic
expression for a capture criterion" that determines whether or
not a disc star with given instantaneous
4D phase-space coordinates may be  in a trapped orbit around the
corotation radius of an imposed spiral pattern (see
Appendix <ref> for a brief review). The physical
interpretation of this capture criterion (given in equation <ref>)
is that the orbital angular momentum of a star is the most important
quantity that determines whether or not that star is in a trapped
orbit.  Indeed, the criterion may be re-stated for kinematically warmed stellar orbits, where such a  star will be in
a trapped orbit provided the instantaneous value of its guiding centre radius lies
within a certain region, the capture region” surrounding the corotation
radius. This capture region is approximately delineated by the closed
contours of the effective potential (Φ_eff, the potential in
the frame that rotates with the spiral pattern) <cit.>.  The capture regions for
three different strengths of spiral perturbation are illustrated in
Figure <ref>, as shaded areas outlined with a
thick, black line (see Table <ref> for the values of the
fixed parameters of the model; the radius of corotation is here chosen to be at 
R_CR=8 kpc).  Contours of the effective potential are shown by
thin lines while the local maxima in the surface density of the spiral
pattern are indicated by thick, dashed (magenta) curves.  The maximum radial
extent of the capture region equals the maximum possible change in
guiding center radius - and hence distance of induced radial migration -
from the given (transient) spiral pattern. This maximum width equals a few
kiloparsec, of order the disc scale-length, for the
parameter values chosen for Figure <ref>.  Most
stars will migrate significantly less than the maximum distance <cit.>.



Circular orbits do not exist in a non-axisymmetric potential due to
non-conservation of angular momentum. We can however consider a star
moving on an orbit along which, at all times, the (time-dependent)
spatial coordinates of the guiding centre equal those of the
star (its orbit has zero random energy). In this case, should the star
be in a trapped orbit, it is physically located within the capture region.  A
star on an orbit with finite random  energy  meets the capture criterion and
is in a trapped orbit provided its guiding centre is
within the capture region, even while the star's spatial coordinates
place it outside the capture region.  The typical amplitude of radial
epicyclic excursions for a population of stars with radial
velocity dispersion σ_R, moving in a potential with local
epicyclic frequency κ, may be approximated by the ratio σ_R / κ. Stars in the local Galactic disc with σ_R
∼ 30 km s^-1 have excursions of order 1 kpc. Taking the finite width of the
capture region into account, for the model of panel (a) Figure <ref>, stars that are members of such a population physically located 2 kpc from
corotation could meet the capture criterion (guiding centre within the capture region). Further, for a flat
rotation curve the epicyclic frequency decreases with radial
coordinate as κ∝ R^-1 so that for given value of
velocity dispersion, the epicyclic excursions are larger in the outer
disc and stars located further from corotation can meet the capture
criterion.

The radial period of trapped orbits for a given spiral
perturbation is longer for larger excursions.[Simple harmonic motion is not a good approximation for trapped orbits <cit.>.]

Our numerical experiments show that the largest
trapped orbits have periods a few times longer than a galactic year at
corotation.  A trapped orbit that has guiding centre excursions of
order ∼ 1 kpc from corotation, in the potential illustrated by
Figure <ref>, panel(a), takes ∼ 8× 10^8 years
to complete a full radial oscillation, whereas a galactic year at
corotation is ∼ 2.2× 10^8 years and the epicyclic period is
∼ 1.6 × 10^8 years.  The radial period of trapped
orbits with the largest excursions places a lower limit on the
timescale for full phase mixing of trapped orbits around corotation. 

Therefore the minimum lifetime for a transient spiral that can
induce radial migration over a significant distance is a few times a
galactic year at corotation.

In this paper, we apply the capture criterion to a series of model
disc galaxies with the aim of determining the (initial)
distribution and fraction of a population of stars in trapped orbits
due to an imposed spiral pattern.[We do not include possible 
additional perturbations to the gravitational potential, such as a
bar or Giant Molecular Clouds, that could scatter stars out of
trapped orbits associated with the spiral of interest.]  While orbital
angular momentum is the physical parameter that is most important in
determining whether or not a star meets the capture criterion, the
orbital angular momentum distribution is far from trivial to determine
from observations. We therefore use velocity dispersion as a proxy.
Our analysis aims to quantify how the velocity dispersion of a
population determines the fraction of disc stars that satisfy
the capture criterion.

This paper is organized as follows: we outline our approach in
<ref>, with the adopted two-dimensional disc potentials described
in <ref>, the analytic forms of the stellar phase-space
distribution functions in <ref>, and derive the resulting
distribution of orbital angular momentum for each model in
<ref>.  We apply our capture criterion to
simple versions of these models in <ref>
in order to focus on how the trapped population depends on
both invariant and radially varying forms of  the stellar velocity
dispersion in <ref> and
<ref>, respectively.  We investigate
how the trapped population depends on other parameters of the model
(such as strength of the assumed spiral) in
<ref>.  We apply our capture criterion to
the full models of the Galactic disc in <ref>.
<ref> places our results in the context of earlier
investigations and <ref> summarises our conclusions.
The Appendix contains a short discussion of the salient points in the
derivation of the analytic capture criterion from Paper I.




§ APPROACH


This section describes our models for the potential of the disc, the stellar kinematics, and the spatial distribution of stars.  We will consider only motion in the equatorial plane of an underlying axisymmetric potential (plus imposed spiral perturbation). Unless otherwise stated, we use the values given in Table <ref> when quantitative calculations are required.






 §.§ Model for the Disc Potential


We assume that the potential governing the motion within an
infinitely thin disc, Φ(R,ϕ) in cylindrical coordinates,
results from an underlying axisymmetric potential, Φ_0(R), plus
a spiral perturbation that is time-dependent in the inertial frame, Φ_1(R,ϕ,t).  We adopt the logarithmic axisymmetric potential
that gives rise to a flat rotation curve,

    Φ_0(R) = v_c^2 ln(R/R_p) ,

where v_c is the (constant) circular velocity and R_p is the characteristic scale of the potential.  We superpose an m-armed spiral density wave[We assume a density wave for two reasons: its form is analytic and the pattern speed is
invariant with radius.] <cit.> of the form 

    Φ_1(R,ϕ) = |Φ_s(R)| cos [m θln(R/R_CR) - m ϕ]

in the (non-inertial) frame rotating  with the spiral pattern speed (Ω_p).  R_CR=v_c/Ω_p is the radius of corotation and θ is the pitch angle of the spiral arm (measured counter-clockwise from a line of constant azimuth).  The zero of the azimuthal coordinate, ϕ, may be chosen arbitrarily and for convenience we chose it to be at the maximum of Φ_eff aligned with  the positive x-axis in Figure <ref>. Further, ϕ increases in the counter-clockwise direction. 

The amplitude of the spiral potential of equation <ref> is given by

    Φ_s(R)=2 π G Σ(R) ϵ_Σk(R),

where Σ(R) is the surface density of the disc, ϵ_Σ is the fractional amplitude in surface density of the spiral pattern, and k(R) is the radial wavenumber of the spiral pattern, defined as:

    k(R) = mR tanθ.

This form  for the spiral potential has a radially dependent amplitude (Φ_s(R)) for any given fractional enhancement in surface density (ϵ_Σ).   Further, the radius of corotation enters explicitly (see equation <ref>); 
we return to these points in 
<ref> below.  



 §.§ Models for the Kinematics of the Axisymmetric Disc


We adopt two different prescriptions for the stellar phase space distribution function, f, to facilitate exploration of the role of the angular momentum distribution. The first assumes simple Gaussians for each component of motion, while the second, taken from <cit.>, produces an asymmetric azimuthal-velocity distribution, with a tail to lower values (matching observations in the local disc). We assume that the phase space distribution is unchanged during the turn-on of the spiral pattern (discussed in the introduction to <ref>).
 


  §.§.§ Gaussian Velocity Distribution




The 4D phase space distribution function for  2D motion in the disc plane can be written in the form <cit.>,

    f(𝐱,𝐯) = Σ(𝐱)  P(𝐯)_𝐱,

where P(𝐯)_𝐱 is the velocity distribution function at coordinate 𝐱 and Σ(𝐱) is the surface density of the disc. We assume that the disc follows an exponential surface 
density profile <cit.>:

    Σ(R)=Σ_0 e^-R/R_d,

with radial scale length R_d.


The velocity distribution function may be approximated by  a 2D Gaussian, which we will denote by P_G(𝐯)_𝐱.  This form cannot reproduce the observed non-Gaussian skew in the azimuthal velocities at a given location in the disc, so  in our models below that adopt the Gaussian, asymmetric drift is incorporated by a simple offset in the mean.


We decompose the velocity (in the inertial frame) into circular velocity plus random velocity: 

    [                            𝐯              = v_R 𝐑̂+v_ϕ ϕ̂;                              =v_ran,R 𝐑̂+(v_c+v_ran,ϕ) ϕ̂;                                               =𝐯_ran+𝐯_c. ]
 
 
The distribution of the radial component of the (random)
velocity is centred on ⟨ v_ran,R⟩=0, with standard
deviation equal to the assumed radial velocity dispersion,
σ_R(𝐱).  The azimuthal velocity distribution is
centred at v_c+⟨ v_ran,ϕ⟩≡ v_c-v_a,
where v_a is the asymmetric drift. As discussed in <cit.> (their sections 4.4.3 and 4.8.2a), for any distribution function that represents a kinematically cool disc (i.e. the random motions are much less than the circular velocity), the asymmetric drift velocity scales approximately as the square of the radial velocity dispersion. Our models are intended to resemble the Milky Way disc and we adopt the following relation, consistent with local observations  and with the (simplified) radial Jeans equation evaluated for the solar location <cit.>:

    v_a= σ_R^2/C,
 
where the scaling constant is set to C=80 km s^-1.

We further simplify the analysis by assuming a fixed ratio of
radial to azimuthal velocity dispersions, equal to that obtained for
the local disc, σ_R/σ_ϕ=0.63 <cit.>
(which is very similar to the value of 1/√(2) expected for
a flat rotation curve; <cit.>).  This allows
us to express the 2D probability distribution for velocity at a given
spatial coordinate in terms of the radial velocity dispersion
alone, as:


    [                                                                            P_G(𝐯)_𝐱                       =12 πσ_R σ_ϕexp{-(v_R-⟨ v_R⟩)^22 σ_R^2-(v_ϕ-⟨ v_ϕ⟩)^22 σ_ϕ^2};                                                                                     = 0.632 πσ_R^2(R)exp{-v_ran,R^22 σ_R^2(R)-(v_ran,ϕ+σ_R^2(R)/C)^22 (σ_R(R)/0.63)^2}, ]


where we have explicitly indicated that  the radial velocity dispersion may vary with radial coordinate. The positive sign in front of the asymmetric drift term of the exponential in the second row of equation <ref> sets the mean azimuthal velocity to slower rotation for kinematically hotter populations.  Populations with different prescriptions for σ_R will also have different orbital angular momentum distributions (and hence different fractions of stars in trapped orbits, for given imposed spiral pattern). This is discussed  in more detail in <ref> below.



The full Gaussian distribution function we adopt in two of the models investigated below  thus has the form

    [                                                                                        f_G(𝐱,𝐯)                                                                                = Σ(𝐱)  P_G(𝐯)_𝐱;                                                                                                 =Σ_0 e^-R/R_d 0.632 πσ_R^2(R)exp{-v_ran,R^22 σ_R^2(R)-(v_ran,ϕ+σ_R^2(R)/C)^22 (σ_R(R)/0.63)^2}. ]




  §.§.§ Distribution Function for a Warmed Disc


The distribution function for an axisymmetric  stellar disc with finite
random motions may be obtained by warming the distribution function
for an initially cold disc (where all stars are on circular
obits). This approach was pioneered by <cit.> and revisited by
<cit.>. In these warmed-disc 
distribution functions, the radial excursions of
stars on non-circular orbits from the inner disc naturally give rise
to asymmetric drift and a low velocity tail in the azimuthal velocity
distribution <cit.>. There are several choices for the form of the warmed-disc distribution function and we  adopt that given as f_new
in <cit.>, which we denote f_D: 

    f_D(E,L_z) = Σ(R_E)√(2)πσ_R^2(R_E)exp{Ω(R_E)[L_z-L_c(R_E)]σ_R^2(R_E)},


where E is orbital energy, which enters the right-handside through the parameter R_E which is the orbital radius for a star in a circular orbit with energy E, Ω(R) is the circular frequency at a given radial coordinate, L_z represents orbital angular momentum (about the z-axis) and L_c(R) is the orbital angular momentum of a star in a circular orbit at radius R.  The surface density generated from f_D is well approximated by an exponential at radii greater than a disc scale length <cit.>.
















  §.§.§ Prescriptions for the Radial Velocity Dispersion Profile


We explore two different types of prescription for the radial velocity
dispersion profile, with two different aims. The first, adopted in
the analyses of sections <ref>
and <ref>, assumes a constant velocity
dispersion, independent of radial coordinate.  This assumption allows
us to estimate how the kinematic temperature affects the
fraction of stars in trapped orbits (after imposition of a spiral
pattern), and particularly to isolate and evaluate the
effects of asymmetric drift (equation <ref>). Since
stellar velocity dispersions are observed to vary with radius, this
first approach is valid only when applied over a limited radial
range. This leads to our second approach, adopting different forms of
the possible variation of velocity dispersion with radius.  We
investigate the results from three specific radial velocity dispersion
profiles (detailed in Table <ref>) in sections
<ref> and
<ref>.





Table <ref> also gives the form of the adopted velocity
distribution function for three fiducial models (denoted Σ, Q
and W).  Model Σ incorporates the form of  σ_R(R) that is appropriate for a  locally isothermal,
self gravitating disc with scale height independent of radius
and with fixed ratio of σ_R/σ_z, namely 
σ_R^2(R)∝Σ(R) <cit.>.  For  an assumed exponential surface density profile, 
σ_R(R)=σ_R,0 e^-R/ R_σ, where R_σ=2 
R_d. As noted in Table <ref>, the radial velocity dispersion is
normalised such that σ_R(R=8 kpc)=35 km s^-1, in
order to approximate solar neighbourhood values.  In this model (Model Σ), we
adopt a Gaussian distribution function, f_G, which includes the
prescription for asymmetric drift given by equation <ref>.




Model Q also uses the Gaussian velocity distribution function (f_G) and the prescription for asymmetric drift (equation <ref>). The velocity dispersion profile, however, is determined by setting the Toomre Q parameter <cit.> for local stellar disc stability to 1.5 at all radii,

    Q≡σ_R(R) κ(R)3.36  G Σ(R)=1.5.


Such a kinematically warm disc is locally stable but may
support global instabilities at all radii
<cit.>. We are interested in modelling a disc
that could have transient instabilities at any radius.  <cit.>
found that transient modes can occur in their N-body
simulations even as the disc is heated to Q>1.5 by spiral
activity.  In a disc with a flat rotation curve,
κ(R)∝ R^-1 so that  σ_R(R) ∝ R 
e^-R/R_d for constant Q. We normalise the model by setting  the surface density at R = 8 kpc to equal 50 M_⊙ pc^-2 (Table <ref>) in order to approximate  the local disc <cit.>.



Model W is constructed under the assumption of a kinematically warmed disc using the  modified Shu distribution function f_D introduced in section <ref>.
<cit.> found that the radial velocity dispersion profile derived from  the second moments of f_D is well approximated by an exponential radial profile such that σ_R(R)=σ_R,0 e^-R/ R_σ, with  R_σ=3 R_d. Unless otherwise noted, the value of σ_R,0 is again normalised such that the radial velocity dispersion σ_R(R=8 kpc)=35 km s^-1.   

The radial profiles for σ_R(R) for each of these three
prescriptions are illustrated in
Figure <ref>.  The curves show Model Σ
(dashed, blue), Model Q (thick, red), and Model W (dot-dashed,
black), normalised as in Table <ref>.  We also show
estimated values for the Milky Way disc from observations of K-giant
stars (green, circles; <cit.> and thin disc stars
from the RAVE survey (magenta, squares) <cit.>.  The
Q=1.5 curve lies below the data for much of the disc, implying
that the adopted normalisation is not well-matched to the Milky
Way. This does not affect the trends that are the main focus of
our investigation.



The capture criterion (equation <ref>) was derived using 
the epicyclic approximation and thus implicitly assumes small
amplitude random motions.  We established the validity of this 
criterion  for any single component of stellar random velocity
up to ∼ 50 km s^-1 <cit.> but we did not test it
above this value.  We therefore use thick/thin lines to distinguish these two regimes in Figure <ref> and subsequent figures. 






 §.§ Orbital Angular Momentum Distributions




The distribution of orbital angular momentum in the disc plays a critical role in determining which stars can be in trapped orbits, and in this section we evaluate this distribution for each of our models[The distribution of orbital angular momentum in the disc may be rearranged at some later time.  However, <cit.> and <cit.> argue that the net change in the orbital angular momentum distribution due to radial migration is nearly zero for any distribution function that has a shallow gradient in orbital angular momentum, ∂ f/∂ L_z, near corotation.]. 


We assume for models Σ and W that a population characterised by radial velocity dispersion σ_R at radius R will have a mean azimuthal velocity ⟨ v_ϕ⟩ that lags the circular orbital velocity by v_a (equation <ref>) and mean guiding centre radius interior to R. 



We calculate the radial distribution of orbital angular momentum by dividing the disc into annuli of width δ R and evaluating  the angular momentum content in each annulus as follows: 

    L_z(R_i)=∫_-∞^∞∫_-∞^∞∫_0^2π∫_R_i^R_i+δ RR   v_ϕ  f(𝐱,𝐯)  R  dR dϕ dv_R dv_ϕ,

where we have assumed that all stars have unit mass (for an individual star, L_z,* = R_* v_ϕ,*).
The orbital angular momentum in each annulus of an exponential disc with a flat rotation curve that is entirely composed of stars in circular orbits is given by,

    L_z,c(R_i)=2π  v_c  ∫_R_i^R_i+δ R Σ(R)  R^2  dR.



The radial distribution of orbital angular momentum, L_z(R_i), for
each model of Table <ref> is shown in Figure <ref>, normalised in each annulus by the corresponding purely circular orbit model (L_z,c(R_i)).  In adopting this  normalisation 
we are implicitly assuming that the 
integral of each distribution function over velocity space gives
a surface density profile that is well approximated by an exponential
(equation <ref>).  The  line types  in Figure <ref> correspond to those in 
Figure <ref>, and again thin lines
represent regions where the radial velocity dispersion
σ_R>50 km s^-1 and underlying assumptions may be expected to break down.




The radial profile for the mean orbital angular momentum goes as
⟨ L_z (R) ⟩∝ R Σ(R)  ⟨ v_ϕ
(R)⟩, where ⟨ v_ϕ (R)⟩ is the mean azimuthal
velocity of the stars at radial coordinate R.
The radial Jeans equations show
that for a given scalelength, populations that have higher random
orbital energy have lower azimuthal streaming velocity (see
<ref> for a brief discussion).  It
follows that the normalised orbital angular momentum distribution for
each model decreases toward the inner regions as the velocity
dispersion increases.

The differences between the shapes of the curves for each orbital angular momentum profile in Figure <ref> can be understood in trems of the differences in each model's radial velocity dispersion profile and radial surface density profile.
For the two models (Model Σ and Model Q) that adopt a Gaussian velocity distribution (P_G(𝐯)_𝐱), the mean angular momentum profile is approximated by ⟨ L_z (R) ⟩∼ R Σ(R)  (v_c-v_a(R)), where v_a∝σ_R^2.
By construction, these two models also have an exponential surface density profile (equation <ref>).  Both of these models also have a radial velocity dispersion profile (σ_R(R)) that goes to zero at large radii, and thus each normalised orbital angular momentum profile converges to unity with increasing radius.   
We also assume an exponential form for the normalisation of the surface density profile for Model W though this is not strictly self-consistent.  An analysis of the robustness of approximating the integral of f_D over velocity space, Σ_D(R), as an exponential with scale length R_d is given by <cit.>.  
We note that at radii R_i/R_d≳ 4, the surface density profile for Σ_D(R) has values greater than those of the exponential surface density profile used in the normalisation and those of models that adopt a Gaussian velocity distribution; this is consistent with the fact that the normalised orbital angular momentum profile for Model W has values greater than unity at R_i/R_d≳ 4.  





§ THE FRACTION OF STARS IN TRAPPED ORBITS


In order to investigate how stellar kinematics affect the fraction of
a population that is in trapped orbits (hereafter denoted by
ℱ), we apply the capture criterion
(equation <ref>) to our models of disc
populations.  In the method described below, we solve numerically for
the fraction of stars in trapped orbits given a particular disc
potential and form for the distribution function. We adopt the
unperturbed, axisymmetric stellar distribution function, then impose a
given spiral perturbation and evaluate for given value of the
phase space coordinates whether or not the capture criterion
(equation <ref>) is met. As we will see, stars
with phase space coordinates not coincident with the corotation
resonance can still be trapped, and the use of the unperturbed
axisymmetric distribution in the calculation of the trapped
fraction needs to be checked. <cit.> investigated the response of both cold and
warm 2D discs to a non-axisymmetric perturbation and found that the
warmer the disc, the less it responded. Azimuthal (their tangential) velocity is the most important phase space
coordinate in our analysis  (our proxy
for orbital angular momentum) and this is less affected than is the
radial velocity.  Bovy found that for typical weak non-axisymmetric
perturbations, such as an elliptical disc or bar (the latter grown either abiabatically or rapidly), the mean tangential
velocity at a given location is perturbed by less than 5%, in a disc
with initial ratio σ_R/v_c < 0.2 (see his Figures 26, 29 and
30). The amplitude of the response was seen  to increase
approximately linearly with the amplitude of the imposed potential. We  expect that while our
results will systematically over-predict the
fraction of stars in trapped orbits, the trends we find with velocity dispersion and location will be valid.

The first step in the evaluation of the trapped fraction is to find the fraction of stars
within each annulus, R_i+δ R, that satisfies the capture
criterion. The numerical integration for a given annulus is carried
out assuming that the fraction is constant across the width of the annulus, and only
one value of the radial coordinate is used, which corresponds to the
inner radius of the annulus[We validated this assumption
provided the width is less than a typical epicyclic excursion.].
 
We integrate the distribution function at R over the remaining phase-space – azimuthal coordinates, ϕ, and velocity space coordinates, v_R and v_ϕ, – with limits of integration set by the region of phase-space that satisfies the capture criterion.  We set absolute limits of integration in velocity space ⟨ v_i ⟩± 100 km s^-1, where i represents the radial or azimuthal components of the velocity vector.
The solution to this integral, divided by the integral of the distribution function over all phase-space at R, is the fraction of stars in trapped orbits within a given annulus. 
We calculate the fraction of stars in trapped orbits for all annuli within R=R_CR±4 kpc for each choice of initial conditions.  We thus find the radial distribution for the fraction of stars in trapped orbits per annulus, ℱ_R≡ℱ(R), where the subscript R" indicates evaluation at the radial position R associated with that annulus.

We perform all integrations using the Wolfram Mathematica v.8.0.4.0 software function  with the parameters  set to 8 significant digits and  set to 8.  We chose the values of these parameters to ensure convergence to a solution within a reasonable amount of time.[This is on order of a half hour of computing time per set of initial conditions with a 4 GHz processor.]  
We set the bounds of integration to be the region of phase-space that satisfies the capture criterion using the  function.

Our primary focus here is to understand trends in the the relationship between the fraction of stars initially in trapped orbits and the stellar velocity dispersion (σ) for the population and so the relative trapped fraction between different models is of most interest.  In the process we explore how the fraction of stars in trapped orbits within a chosen radial range, 

    ℱ_Δ R≡∫_Δ Rℱ(R)  dR ,

depends on the adopted functional form of the distribution function and the parameter values of the spiral potential.[Equation <ref> implicitly includes an integration of the chosen distribution function over all phase-space, excepting radial position, as described at the beginning of <ref>.]
The subscript Δ R" indicates that the fraction of stars in trapped orbits is evaluated by integrating over a specific radial range, Δ R, of annuli, where the value for ℱ_i within each annulus has been evaluated over the non-axisymmetric azimuthal distribution of stars in trapped orbits (as described above).  We henceforth refer to the fraction of stars in trapped orbits within a chosen radial range (ℱ_Δ R) as the integrated fraction.
We do not evaluate the time evolution of
the integrated fraction, and therefore refer to the fraction of stars that meets
the capture criterion for a given model as the initial"
fraction.   In <ref>, we argue that the initial fraction over-predicts the fraction of stars that could migrate radially.

For the entirety of this section we adopt either a radially fixed velocity dispersion profile and Gaussian distribution function (f_G(𝐱,𝐯)) or our simplest fiducial model, Model Σ.  In Section <ref>, we use insights gained from these explorations to investigate the full fiducial models, Models Σ, Q, and W, of the galactic disc.






 §.§ Integrated Fraction for a Fixed Radial Velocity Dispersion and Gaussian Distribution Function


The fraction of stars in a given disc population that meets the capture criterion
at a given spatial coordinate depends on the population's velocity dispersion[This statement should not be confused
with the fact that whether or not an individual disc
star is in a trapped orbit is largely independent of its random
orbital energy and determined primarily by its orbital angular momentum (or circular frequency)<cit.>.]. In this subsection we use a 
simple model to gain physical insight into the form of this dependence.
We adopt the Gaussian phase space distribution function
f_G(𝐱,𝐯) (equation <ref>, described in
<ref>), a radial velocity dispersion (σ_R) that is
constant within the disc and a fixed ratio between the radial and
azimuthal velocity dispersions, σ_ϕ/σ_R.  The disc and spiral pattern are assumed to have the parameter values given in Table <ref>, with R_CR=8 kpc.






  §.§.§ Radial distribution of the fraction of stars in trapped orbits


Figure <ref> shows the fraction
of stars in trapped orbits at a range of radial coordinates for a
given radial velocity dispersion, σ_R, ranging between 5-80 km s^-1, in
5 km s^-1 increments.   We devote the rest of this
subsection to identifying the physics that determines the shapes of
these curves.  The intuition developed will be useful when we investigate 
more complex models later in this paper.






In the limiting case that σ_R → 0 and all orbits
approach circular (in an axisymmetric potential), the radial distribution of
the fraction of stars in trapped orbits saturates at
100% within the capture region and goes to zero at coordinates
outside the capture region.  An approximation for the integrated
fraction (ℱ_Δ R) in this limit can be obtained from the ratio of the area of the  capture region, A_cap, to 
the area of the annulus, A_ann, that spans the maximum radial range
of the capture region. For low-amplitude spiral arms (ϵ_Σ≲ 0.3) that have a low pitch angle
(θ≲ 25^∘) the capture region follows closely the closed contours around the maxima in
the effective potential (as seen 
in Figure <ref>(b)) and these in turn are well approximated by a
sinusoidal function in the azimuthal direction (see
equation <ref>).  Under
these conditions the area of the
capture region may be evaluated as:

    A_cap = ∫_0^2π∫_R_CR -(R_CR-R_min)|cosϕ |^R_CR +(R_max-R_CR)|cosϕ | R dR dϕ,

where R_min and R_max denote the minimum and maximum
radial coordinates of the capture region.  Making the reasonable
assumption that the capture region is symmetric around  corotation, and
being careful to note that the limits of integration involve |cosϕ|, it is straightforward to show that 
A_cap=4 R_CR(R_max-R_min).  The area of the
annulus enclosing the capture region (A_ann) is A_ann =
2π R_CR(R_max-R_min). The integrated fraction of stars in the
annulus containing the capture region is therefore
ℱ_Δ R=2/π≈ 64%.  Values higher than 
 2/π can be obtained if the annulus over which the trapped fraction is integrated is narrower than the capture region, with the integration fraction approaching 100% around corotation, where the capture region spans all azimuthal
coordinates. These limiting cases will be used to inform our interpretation of results that
appear later in this paper.




The rms  maximum radial excursion from the guiding centre radius, due to epicyclic motion, for a population with radial velocity dispersion σ_R is given  by

    X(R) = σ_R(R)√(2) κ(R)


Thus in a disc with constant radial velocity dispersion and a flat rotation curve, the rms maximum radial epicyclic excursions scale as X ∝ R. This typical maximum epicyclic excursion is a measure of how far from the limit of the capture region a star can be physically and still have its guiding centre within the capture region and be in a trapped orbit.  

The curve in Figure <ref>
for the lowest velocity dispersion (σ_R = 5 km s^-1) peaks
close to the corotation radius (8 kpc), and 
the fraction of stars in trapped orbits in annuli away from corotation
approximately equals the fraction of azimuthal coordinates that are
within the capture region.   Populations of stars with
higher values of their velocity dispersion, on  more
eccentric orbits, have a greater range of radial positions outside the capture
region and therefore the distribution of stars in trapped orbits spreads over a greater
radial range as seen in  Figure <ref>.
High velocity dispersion populations can have a distribution of stars in trapped orbits that extends beyond the inner and outer Lindblad resonances (radii at which κ=± m(Ω_p-Ω_c) – marked with long-dashed in Figure <ref>) where stars are likely to be scattered out of trapped orbits.

The location of the peak in the trapped fraction also depends on
velocity dispersion, shifting to larger galactocentric radii for
kinematically hotter populations. This reflects the shift in mean
azimuthal velocity, represented by our simple asymmetric drift formula
(equation <ref>).  To understand this, first consider how the
contribution to the integrated fraction from a stellar
population at given coordinates 𝐱 depends on velocity
dispersion. This is illustrated in
Figure <ref>, which shows the
distribution of random velocities in the azimuthal direction for stars
observed at coordinates R=8.5 kpc, ϕ=0 (just outside
corotation, between the spiral arms), and with v_ran,R=0. The different curves are for radial velocity dispersions in the range 5-80 km s^-1, in
5 km s^-1 increments. 
The peak of the distribution of v_ran,ϕ and of 
the corresponding distribution of guiding centre radii, R_L, shift to lower values with higher radial velocity
dispersion.  The yellow (shaded) region indicates those values of v_ran,ϕ
(and R_L) that meet the capture criterion.  




The radial coordinate of the peak in the radial distribution of the fraction of stars in
trapped orbits (R_peak – in Figure <ref>) corresponds to the radius at which
the mean guiding centre radius for the stellar population of interest equals the
radius of corotation.
The radius of corotation is the only radius at which all azimuthal coordinates are within the capture region, modulo a slight radial broadening for very strong spiral patterns (compare panels (b) and (c) in Figure <ref>).  

The mean guiding centre radius (⟨ R_L ⟩) of stars
observed at coordinate R reflects the mean angular momentum of the
stars, which in our prescription is set by the velocity dispersion,
through the asymmetric drift equation (equation <ref>). 
Thus, stellar populations with velocity dispersion approaching zero (σ_R→ 0), where the radial position of each star approaches its guiding centre radius (R→ R_L), have R_peak→ R_CR.
R_peak increases (moves to larger distances from the galactic
centre) with increasing velocity dispersion.  We show the
dependence of R_peak on the radial velocity dispersion in
Figure <ref>; the thick (black) curve represents
the offset R_peak-R_CR for each value of the radial
velocity dispersion shown in
Figure <ref>.







These physical arguments suggest that the value of R_peak can
be determined by finding the radius at which the mean guiding centre radius equals the
radius of corotation. We then have:

    ⟨ R_L ⟩=R ⟨
    v_ϕ⟩/v_c

so that 

    R_peak ⟨ v_ϕ⟩=R_CR v_c
 
and 

    R_peak-R_CR = R_CR(v_cv_c-v_a-1 ).











The predicted relation from Equation <ref> is
  shown in Figure <ref> (dashed, red curve)
  together with the results from our models (solid, black curves). The
  relation from Equation <ref> is a good match to the
  models, up to some critical value for the velocity dispersion,
  above which the offset R_peak-R_CR reaches a
  plateau. This maximum value for the offset is obtained when the
  radial coordinate for the population of stars that has a mean
  guiding centre radius equal to the radius of corotation is greater
  than, or equal to, the maximum radius of the capture region
  (R_max).  Beyond this limit, stars in the 
peak
  of the velocity distributions no longer meet the capture
  criterion and  the contribution to the trapped fraction comes from stars in the broad, low-amplitude wings. Thus this critical value of the velocity dispersion  marks the boundary
  above which the dominant contribution to the fraction of stars in
  trapped orbits changes from stars that have both radial positions
  and guiding centre radii inside the capture region to stars with
  guiding centre radii only inside the capture region.  However since
  the capture region does not have constant radial width (see Figure <ref>), the
  transition between these two regimes is not abrupt.

An expression for this critical value of the radial velocity
dispersion, σ_crit, may be obtained by considering 
the  stellar population located at
the maximum radius of the capture region and requiring that the mean angular momentum (per unit mass) of this population, given by   ⟨ L_z⟩ = R_max ⟨
v_ϕ⟩,  be equal to that of a star in a circular
orbit at corotation (i.e. ⟨ L_z⟩ =R_CR v_c).   Given this
requirement and our prescription for asymmetric drift
(equation <ref>), the critical velocity dispersion may be expressed as:

    σ_crit = √(C v_c(1-R_CRR_max))

The value of σ_crit for the parameter values of the model in  Figure <ref> is indicated by the  thin, vertical line in that figure.





  §.§.§ Definition of the radial range of evaluation


The fraction of stars initially in trapped orbits varies with 
radius and it is important  to quantify  how the integrated
fraction (ℱ_Δ R) depends upon the radial
range over which it is evaluated.  We define this radial
range, Δ R, with two different approaches, the first a fixed physical width and the second depending on the shape of the distribution of the trapped fraction. 
Two of the measures of the trapped fraction, denoted ℱ_1 and ℱ_2, are
evaluated using annuli centred at the corotation radius and of fixed width  Δ R_1 = 1 kpc and 
Δ R_2 = 2 kpc, respectively.  We also evaluate three measures that
do not have a fixed centre, rather these radial ranges for evaluation
are defined in terms of 
the radial distribution of stars in trapped orbits.  For these,
denoted by ℱ_5%, ℱ_25%, and ℱ_FWHM, the requirement is that in every
annulus (each of width δ R, given in Table <ref>) within the
radial range of evaluation, the fraction of stars in trapped orbits must be greater than
5% (defining the radial range Δ R_5%), 25% (Δ R_25%), and half
the maximum fraction captured (Δ R_FWHM), respectively.



The widths of the above  radial ranges of interest are illustrated
in Figure <ref>, for 
the model and parameter values of 
Figure <ref>. 
For low
values of the velocity dispersion, R_peak→ R_CR and the shape-dependent radial ranges approach an annulus centred on  corotation, as is prescribed for the
fixed measures Δ R_1 and Δ R_2. The variable widths generally are larger than the fixed widths, as seen in the figure.  Radial ranges narrower than the capture region will give higher integrated fractions than wider radial ranges and indeed can lead to higher values than the ∼ 64% we estimated for widths equal to that of the capture region. Again, the widest radial ranges at high velocity dispersions can be comparable to the distance between the inner and outer Lindblad resonances in
Figure <ref> and may include stars that are likely to be scattered out of trapped orbits.



An illustration of these various measures of the radial range of integration is given in Figure <ref>, applied to the curve of the distribution of the trapped  fraction for populations with σ_R={15,35,55} km s^-1, from  Figure <ref>.  For populations with high radial velocity dispersion (e.g., σ_R=55 km s^-1 in Figure <ref>), the peak of the distribution of stars in trapped orbits may not be included in evaluations of the integrated fraction that use a fixed radial range (Δ R_1,2). 






  §.§.§ Dependence of the integrated fraction on velocity dispersion


Figure <ref> shows the integrated
fractions ℱ_1, ℱ_2, ℱ_5%,
ℱ_25%, and ℱ_FWHM, together with the best-fit linear fits to the trends as a function of velocity dispersion, for the  model parameters of  this subsection.  The open symbols and thin lines denote values of the radial velocity dispersion above 50 km s^-1, and the 
thin,
vertical line indicates the critical velocity dispersion,
σ_crit, above which one might expect a change in behaviour.



As seen in Figure <ref>, for velocity
dispersions below around 30 km s^-1, the radial ranges used to
calculate the trapped fractions are narrower than the capture region,
with the exception of R_5%. This explains why the various measures
of the trapped fraction are higher than the 64% that was derived
earlier for the radial extent equal to the capture region. The values
of the measure ℱ_5% follows our expectations. This
measure includes a larger fraction of the disc and should be more
representative, modulo the caution that at large values of the
velocity dispersion the radial range becomes so wide that interactions
with the Lindblad resonances cannot be ignored. Indeed it is for this reason that we do not make an evaluation of the trapped fraction by integrating over the entire disc, but rather limit the evaluation to the maximum radial range provided by Δ R_5% and caution that ℱ_5% may include a significant population that is scattered out of trapped orbits, particularly at high velocity dispersions.




All measures of the trapped fraction show  a clear, approximately linear, decrease with increasing  velocity dispersion of the stellar population.
The linear[We also explored exponential,
polynomial, and Gaussian functional forms, but there is no  compelling physical reason to include these more complex relationships.] fits shown in the figure used the function  from Wolfram
Mathematica's built-in library[We set the fit method" to Automatic",
thus allowing Mathematica to select from its library to
minimize fitting residuals.]  The values of the slope and zeropoint of the following linear fit are given in Table <ref>:


    ℱ_Δ R(σ_R) = ξ_μσ_R +ξ_β ,



We expect the values of the fitting constants, ξ_μ and
ξ_β, to depend on the underlying parameters such as the radius of 
corotation and the amplitude of the spiral potential, and explore this in section <ref>.













 §.§ Integrated Fraction for Radially Dependent Velocity Dispersion
 

Our assumption in the last section that the radial velocity dispersion is constant with radius is reasonable provided that only a relatively narrow range of radius is being considered.   We now explore how the trends in the
integrated fraction shown in
<ref> are modified when a
 radially dependent velocity dispersion profile
(σ_R=σ_R(R)) is assumed.




Throughout this subsection we assume Model Σ, which
is summarized in Table <ref> and uses the disc
and spiral potentials described in <ref> with
parameter values given by Table <ref>, unless otherwise
stated. This model assumes an exponential radial velocity
dispersion profile, with scale length twice that of the assumed
exponential disc. We explore a range of normalisations at corotation, 
σ_R(R=8 kpc) = { 5, 80} km s^-1 in
5 km s^-1 intervals.  The dependence of the typical (rms)
amplitude of epicyclic excursions (equation <ref>) on both
epicyclic frequency (κ(R), equation <ref>) and the
local value for the radial velocity dispersion (σ_R (R)) is
taken into account in the evaluation of the trapped fraction.


The results are illustrated in
Figure <ref> where individual
panels are analogues of
Figures <ref>-<ref>)
and show (a) the radial distribution of the initial fraction of
stars in trapped orbits, ℱ_R(R); (b) the
distance between the peak of ℱ_R(R) and the radius of
corotation (R_peak-R_CR), as a function of velocity
dispersion normalisation; (c) the various radial ranges (Δ R)
within which the integrated fraction is evaluated, and (d) the
integrated fraction (ℱ_Δ R).



The main difference between the curves for ℱ_R in
Figure <ref>(a) and
Figures <ref> is a steeper
decline at larger galactocentric radii in the case of the radially
varying velocity dispersion, particularly for larger values of the
normalisation at coorotation (8 kpc).  This reflects the fact that the
local velocity dispersion in the outer disc is lower compared to the
invariant case, so that the typical epicyclic excursions are lower,
such that stars observed in the outer disc are less likely to have
guiding centres within the capture region around corotation. The
trapped fraction is low in these regions, far from corotation, so the
differences are not large in terms of parameters such as the
integrated fraction, Figure <ref>(d). The widths of the various radial ranges (panel (c)) are also narrower, compared to the case of constant velocity dispersion, again reflecting the narrower distributions in panel (a).  Other than these minor quantitative differences, the qualitative results agree in showing a  decline of trapped fraction with increasing velocity dispersion. 






 §.§ Other Functional Dependencies for the Integrated Fraction


In this section, we assume the same model used in
<ref>, where we adopt a Gaussian
velocity distribution at each coordinate (f_G, described in
<ref>), radially fixed radial velocity dispersion
profile (σ_R=35 km s^-1), and exponential surface density
profile (Σ(R), equation <ref>).
Unless explicitly stated, model parameter values are given in
Table <ref>.

The integrated fraction of stars initially in trapped orbits
(ℱ_Δ R) depends on the size and shape of the
capture region, which depends on the spiral pattern speed (Ω_p)
and the underlying potential of the disc and spiral
(Φ(R,ϕ)=Φ_0(R)+Φ_1(R,ϕ)) <cit.>.  For a
given pattern speed, the integrated fraction increases with
increasing amplitude of the perturbation to the potential due to the spiral pattern (|Φ_s|_CR), since the width of the
capture region is larger for stronger spiral strengths (see
Appendix <ref>).  In our model, the strength of the spiral perturbation 
(equation <ref>) is written in terms of  the radius of
corotation (R_CR), spiral pitch angle (θ), number of spiral
arms (m), the assumed surface density profile (Σ(R)),
and the fractional surface density enhancement of the spiral potential
(ϵ_Σ) (see <ref>).  The model assumes a flat rotation curve of given circular velocity, so that `corotation radius' is a proxy for `pattern speed'.
Figure <ref>  shows how  the maximum
width of the capture region (R_max-R_min) in our model varies with increasing  fractional surface density of the spiral pattern
(ϵ_Σ) for fixed corotation radius (left panel) and with increasing  radius of corotation (R_CR) for fixed fractional surface density (right panel). 



The curve in the lefthand panel of Figure <ref> is
approximately a parabola, so that the (maximum) width of the capture
region scales as ϵ_Σ ^1/2. For all other parameters
fixed (as here), ϵ_Σ is linearly proportional to the
amplitude of the spiral potential (equation <ref>). Previous work <cit.>,  established the expectation that the maximum radial range
of a trapped orbit would scale as the square root of the strength of
the imposed perturbation. The dependence seen in the figure then would
imply that the width of the capture region scale as the maximum radial
range of a trapped orbit. Since stars can be in trapped orbits while their physical coordinates lie outside the capture region, this correspondence is expected to be only approximate. 

The non-monotonic behaviour seen in the righthand panel reflects
how, assuming the same fractional surface density 
(ϵ_Σ), the radial range of the capture region does not
scale as √(|Φ_s|_CR) as one might naively expect.[Our 
prescription for the amplitude of the spiral pattern (Φ_s(R))
depends on the (exponential) surface density profile (Σ(R)) and the wave number (k(R),
equation <ref>), so that the peak amplitude goes as
|Φ_s|_CR∝ R_CR e^-R_CR/R_d (see
<ref>) with maximum strength when R_CR=R_d.  ]
Figure <ref> illustrates how, with increasing
spiral amplitude, not only does the radial range of the capture region broaden, but also  
its shape acquires a skew distortion from the simple sinusoidal model
assumed in the derivation of ℱ_Δ
R(σ_R→ 0) in <ref>
<cit.>.  This skew becomes more prominent for
higher amplitude spiral perturbations.
Thus for lower values of the corotation radius, where the surface density profile (Σ(R),
equation <ref>) is exponentially greater, the resulting radial
range for the capture region is lower than the naive
expectation, where the least skew distortion per spiral strength appears to occur at ∼ 3R_d
in our models.





























We turn next to investigations of how the trapped fraction depends on these parameters of the model potential.





  §.§.§ Integrated Fraction as a Function of  Spiral Strength


We here use the value for the fractional surface density
(ϵ_Σ) as a proxy for the amplitude of the spiral
potential at corotation (|Φ_s|_CR) when all other parameters
are held constant.  In Figure <ref>, we
use the same five values for the fractional surface density
(ϵ_Σ) as in Figure <ref> and a radially
invariant radial velocity dispersion
(σ_R=35 km s^-1).  Panel (d) of
Figure <ref> shows that the
integrated fraction of stars in trapped orbits (ℱ_Δ
R) increases smoothly with increasing fractional
surface density (ϵ_Σ).  The shape of the curves in
Figure <ref>(d) are remarkably similar
to shape of the curve in the left hand side of
Figure <ref>, indicating 
that the integrated fraction scales with the the maximum width of the
capture region (or area, for given value of R_CR).    A
similar behaviour is expected for the dependence of the integrated fraction
on any other parameter that enters linearly in the expression for
spiral strength (equation <ref>), such as number
of arms (m).  

There is a 0.1 kpc decrease in the offset between corotation and the peak
of the radial trapped fraction profile from ϵ_Σ=0.3 to ϵ_Σ=0.4,
as shown in panel (b) of Figure <ref>.  This could arise from
the relative importance of the skew in the shape of the capture region with increasing amplitude
for the perturbing potential (see discussion in <ref>).
However, it is difficult to draw significant conclusions about this shift since changes in the 
values for R_peak-R_CR between models are less than or on order of δ R (Table <ref>).




There is a clear trend that the integrated fraction of stars in trapped orbits (ℱ_Δ R) increases with increasing spiral amplitude at corotation (|Φ_s|_CR).  The fitting coefficients in equation <ref> (ξ_μ,β) also depend on spiral strength, but we have found no simple scaling or transformation. 



  §.§.§ Integrated Trapped Fraction as a Function of Pattern Speed


In <ref> we demonstrated that the
integrated fraction of stars in trapped orbits (ℱ_Δ
R) depends on the amplitude of the spiral perturbation to the potential at  corotation (|Φ_s|_CR)
since the maximum width  of the capture region increases with increasing
strength of the spiral perturbation.  Our prescription for the amplitude of the
spiral pattern, (Φ_s(R), equation <ref>)
depends on the both the disc  surface density profile, which we have chosen to be an exponential 
(Σ(R), equation <ref>) and the
wave number (k(R), equation <ref>), such that  the
peak amplitude scales as |Φ_s|_CR∝ R_CR e^-R_CR/R_d
(see <ref>).  In the introduction to this section
(<ref>) and the right-hand panel of
Figure <ref> we showed that the maximum width of the
capture region (R_max-R_min) scales non-monotonically with the
radius of corotation .  In this subsection we explore how and why
the integrated fraction depends on the radius of corotation
(R_CR), and its corresponding spiral pattern speed (Ω_p =
v_c/R_CR), including the size of the capture region.

The size of epicyclic excursions, as well as the position and shape of the capture region, affect the radial distribution of stars in trapped orbits (ℱ_R).
In Figure <ref>, we set σ_R=35 km s^-1 at all radii, let the phase space distribution be given by f_G(𝐱,𝐯) (<ref>), and calculate the various measures of trapped fraction (ℱ_R) for pattern speeds corresponding to radii of corotation between 4-15 kpc, in 1 kpc intervals.





For fixed radial velocity dispersion (σ_R) and flat
rotation curve, the typical stellar radial epicyclic excursions
increase linearly with galactocentric radius, X∝ R (see
equation <ref>).  The distribution of stars in trapped
orbits (ℱ_R) therefore depends on the radius of
corotation (R_CR) since both the maximum amplitude of epicyclic
excursions (X) and the width of the capture region
(|R_max-R_CR|≈ |R_CR-R_min|) set the maximum distance
from corotation for stars in trapped orbits and R_CR sets the area of the capture region (see discussion following
equation <ref>).  The maximum range in radial coordinate for stars in trapped orbits
is illustrated in panel (a) of
Figure <ref> and best quantified by
Δ R_5% (long-dashed, orange) in panel (c).  The shape of the curve for
Δ R_5% in Figure <ref>(c) is
consistent with that for the width of the capture region as a function of
corotation (see Figure <ref>) combined with the linear increase in typical  maximum radial epicyclic excursions
(X∝ R). 

Equation <ref> predicts that the value for R_peak-R_CR scales as R_CR for fixed velocity dispersion (a consequence of epicyclic excursions scaling with radius in this regime).  Panel (b) of Figure <ref> illustrates this point.  The curve in this plot is not smooth (and also in panel (b) of Figure <ref>).  As stated in <ref>, this could be due to the relative importance of the
skew in shape of the capture region with increasing spiral amplitude but it is unclear since changes in 
the values for R_peak-R_CR between models are less than or on order of δ R (Table <ref>).

The trend is that the integrated fraction (ℱ_Δ
R) generally decreases with increasing radius of corotation, equivalent to decreasing pattern speed..  All
evaluations of ℱ_Δ R will naturally decrease in terms of the maximum value, since
ℱ_R broadens with
increasing R_CR (as also discussed  in
<ref>). Should the offset
(R_peak-R_CR) be large enough for the peak to be beyond the annulus over which the integrated fraction is evaluated, as can happen for fixed annular widths such as Δ
R_1,2, the integrated fraction in that case (ℱ_Δ R) decreases
more rapidly with increasing corotation radius than when
ℱ_Δ R is evaluated using non-fixed radial ranges 
(i.e. Δ R_5%,25%,FWHM).






§ INTEGRATED TRAPPED FRACTION IN MODELS OF THE GALACTIC DISC


In <ref>, we established that the
value of the integrated fraction depends strongly  on the value of the radial
velocity dispersion at corotation (σ_R(R_CR)). 
We also found  that the integrated
fraction depends on the area and location of the capture region, the size
of epicyclic excursions, and the spiral strength and pattern speed.
These analyses in hand, we are equipped to interpret how our adopted forms for the stellar distribution function, including 
particular prescriptions for the radially dependent radial velocity
dispersion profile, affect the derived behaviours of the integrated fraction
over the disc.


Our results for the three fiducial models outlined in
Table <ref> are shown in
Figure <ref>, for different
values of the corotation radius such that R_CR varies between
4-15 kpc, in 1 kpc intervals.  Each model adopts the parameter
values in Table <ref>, where the normalisation of the
radial velocity dispersion profile (σ_R(R)) for each
model is given in Table <ref>.  In this section, we
will only consider regions of the disc where σ_R ≤
50 km s^-1.



For all values of the corotation radius, we find that the
trends in the width of the unconstrained radial ranges for
evaluation (Δ R_5%, Δ R_25%, and Δ
R_FWHM) and their corresponding integrated fractions
(ℱ_5%, ℱ_25%, and ℱ_FWHM),
primarily depend on both the area of the capture region
and the typical amplitude of epicyclic excursions
for stars in trapped orbits (parameterised by the rms
maximum amplitude, X).  These in turn depend on the value of
R_CR.  The maximum radial width of the capture region, which
closely scales in amplitude with its area, is shown as a function of
corotation in the right hand panel of Figure <ref>.
The width of the capture region depends on the adopted
underlying potential <cit.>.  The curve in the right
hand panel of Figure <ref> is applicable to
all of our fiducial models and is consistent with the maximum radial
width for the capture region roughly scaling with the radius of
corotation as (R_max-R_min)∝ R_CR e^-R_CR/3R_d.
The area of the capture region influences the
radial width of the distribution of stars in trapped
orbits and, more importantly, the local surface density or
number of stars in trapped orbits.  Taking into account
that the shape and area of the capture region for a given radius of
corotation is the same for all fiducial models (assuming fixed
remaining parameters of the spiral) since they assume the same
underlying potential, we can explore how the kinematics of each model
affects the integrated fraction of stars in trapped orbits.

The radial dependence of the velocity dispersion affects the integrated 
fraction in two ways.  First, the radial velocity dispersion sets the amplitude of radial
epicyclic excursions, X∝σ_R(R)/κ(R)
(equation <ref>).  In turn, the sizes of epicyclic excursions affects the radial width of
the distribution of stars in trapped orbits (ℱ_R) since these determine the typical difference between a star's physical radial coordinate and that of its guiding centre.
Second, the velocity dispersion sets the amplitude for the offset R_peak-R_CR 
(panels (b))
through its dependence on the asymmetric drift (see
eqn <ref> earlier). 
The behavior of the  curves for each model in
Figure <ref>, which illustrate how the velocity dispersion underlies the shape of the 
orbital angular momentum distribution, is related to this interdependence.  The angular momenta decrease toward the galactic
centre, compared to the case of all stars on circular orbits, as velocity dispersion increases.  The overall consequence is that a lower fraction of stars near the galactic centre meets the capture
criterion for each model.
The values for unconstrained measures of the integrated fraction 
(ℱ_Δ R=ℱ_5%,25%,FWHM) quantify the shape of the distribution
of stars in trapped orbits, where the range in radii
for the various measures ℱ_R is
closely approximated by that given by Δ R_5%
(long-dashed, orange curve in panels (c)). 
The assumed radial velocity dispersion profile (σ_R(R)) is 
therefore integral to understanding the integrated
fraction ℱ_Δ R  for each model since it sets the 
radial dependence of the radial offset, R_peak-R_CR, and the shape (measured by Δ R) of the distribution of stars in trapped orbits.  
The following is a more detailed exploration of the distributions shown in Figure <ref> for each model.

The radial velocity dispersion profile for Model W
(σ_R(R)∝ e^-R/3R_d) is relatively shallow
(Figure <ref>, dot-dashed, black) and leads to
an approximately constant value for R_peak-R_CR as
the value for the corotation varies 
(Figure <ref>(b), last column).  
The curve for Δ R_5% peaks at approximately 3 R_d where its value is nearly
2 kpc greater than its minimum value.  The shape of
this curve arises from two competing effects.  The first
is the dependence of the area of the capture region on
the value of R_CR, as described above and in the introduction to
<ref>.  Second, the amplitude of radial
epicyclic excursions in Model W goes as
X_W∝ R e^-R/3R_d.
Thus, in Model W, the variation of the width of
ℱ_R from epicyclic orbits equals the radial
dependence for the total number of stars in trapped orbits and
these two effects act in opposing
fashions.  The final result of this balance is that the value
of the integrated fraction for ℱ_5% (and other
unconstrained measures of ℱ_Δ R) is nearly constant
with radius.  Fixed measures for ℱ_Δ R
(i.e. ℱ_1,2) follow (R_max-R_min) and do
not have a significant radial dependence on asymmetric drift.

The radial velocity dispersion profile for Model Q approaches zero at
large radii.  Thus with increasing values for the corotation
radius, the offset of the peak of the distribution of the trapped
fraction (R_peak-R_CR) goes to zero. Further, the
unconstrained radial ranges converge to the width of the
capture region (as most of the stars in trapped orbits are
physically in the capture region), the maximum
values for ℱ_R approach 100% (as most of the stars
near corotation are in trapped orbits), and
ℱ_5% approaches ∼ 64% (the result from
equation (16) in section 3.1.1 earlier).  The amplitude
of radial epicyclic excursions for Model Q goes as X_Q∝
R^2 e^-R/R_d, which peaks closer to the galactic centre
(at 2R_d) than the expression for the radial range of the
capture region at (∼ 3R_d).  Should the offset R_peak-R_CR be
constant, this would have the effect of broadening the distribution of
the trapped fraction compared to Model W.  However, the offset
R_peak-R_CR increases with decreasing corotation radii, reaching  a value of nearly
half the width of the capture region.  
The overall consequence is that
a lower fraction of stars near the galactic centre meets the capture
criterion, compared to Model W, and the integrated fraction for Model Q
thus decreases toward the galactic centre.

The trends in the curves for Model Σ (Figure <ref>, middle column) are similar to those for
Model Q.  Both these models assume the same form for the distribution function (f_G), but different radial velocity dispersion profiles.  The normalisation for the radial velocity dispersion
profile for Model Σ sets the local value for σ_R to
greater values than in Model Q at all radii.  Model Σ also has the
steepest (declining with radius) radial velocity dispersion profile of all models and
consequently has the lowest fraction of stars that meet the capture
criterion toward the galactic centre (see also the dashed, blue curve
in Figure <ref>).  The radial velocity dispersion
profile for Model Q was selected in order to approximate a disk that
could support gravitational instabilities at all radii.  Interestingly, this also produces a higher
integrated fraction at all corotation radii.

For all models with radially dependent radial velocity dispersion
profiles, the value for the integrated fraction
(ℱ_Δ R) for any given radial range for
evaluation (Δ R) is rather constant (±∼ 10%) as a
function of corotation radius.  
This nearly constant value for the integrated fraction
primarily arises from (1) the identical radial dependences for the
radial range of the capture region (Figure <ref>) for each model and (2) the similar distribution of 
orbital angular momenta for stars with guiding centres within the capture region – where
 the associated measurables are the size of epicyclic 
excursions (X(R)) and the offset offset R_peak-R_CR – due to the 
assumed kinematics for each model.
As shown in <ref>,
the integrated fraction decreases with increasing value for the
normalisation of the velocity dispersion profile.



§ DISCUSSION




 §.§ Integrated Trapped Fraction as an Upper Limit


The initial distribution of stars in trapped orbits derived
above should be interpreted as an upper limit, for the following
reasons.  First, the parameter Λ_nc,2(t), used
to evaluate whether or not a star is in a trapped orbit, is
explicitly a time-dependent quantity (equation <ref>) and we showed in Paper I that stars which initially satisfy the
capture criterion could be scattered[We define scattering" as an event where a star' s change in orbital angular
momentum is correlated with a change in random orbital energy,
distinct from radial migration.] out of a trapped orbit before the torques from the spiral perturbation had time to change significantly that star's orbital angular momentum.
Scattering events could affect the fraction of stars in trapped orbits
over time (see discussion in Paper I, 3.3).  This is an especially
important point for stars in trapped orbits that either (1) have
large radial excursions from the guiding centre, since such stars
sample a larger fraction of the underlying disc potential and
are therefore are more likely to scatter as they encounter
inhomogeneities in the disc,[A star in a trapped orbit
that has large excursions from its guiding centre

could have a coordinate position that is distant from the capture region.  In a realistic disc, such a trajectory would likely lead to scattering via interactions with inhomogeneities in the disc potential that are associated with objects such as dark matter substructure and giant molecular clouds.  When modelling radial migration in the disc, one should also take care that the choice for Δ R is less than the radial distance between spiral arms (here λ(R)=2π/k(R)), where the radial wavenumber k(R) (equation <ref>) is proportional to the number of spiral arms (m).] or (2) meet a resonant criterion in addition to the corotation resonance – e.g.  the path for the guiding centre radius crosses an ultra-harmonic Lindblad resonance. 

Indeed, <cit.> observed that a sufficiently long-lived
spiral perturbation caused not only the pattern of angular momentum
changes that correspond to trapped `horseshoe' orbits, but also a later pattern of opposite angular momentum changes for
stars close to the corotation radius, consistent with scattering
processes.

Although  it is generally accepted that discs are kinematically heated over time, the mechanisms and their relative importance aren't entirely clear.
<cit.> argued that a rapidly growing spiral perturbation would cause a non-adiabatic response in the stellar disc over a broad radial range and a second order change in the orbital angular momentum distribution.  
In a high resolution follow up study using a pure stellar disk, <cit.> showed that even long lived material spiral arms  kinematically heat the disk and they further supported this claim with analytic arguments.  Indeed,  based their analytic argument for  heating rates on those from
GMCs and other small scale fluctuations in the galactic potential <cit.>.
In any case, we have shown that a kinematically heated population of stars has a lower fraction of stars in trapped orbits.  This implies that such heating by the spiral would over time lead to fewer stars being captured in trapped orbits than would be expected from the initial integrated fraction.

Our analysis assumes that all orbits are constrained to a 2D
disc.  This approximation is valid for stellar orbits that are
nearly circular since the vertical and radial actions are separable
<cit.>.[Indeed, we expect the capture
criterion to be less robust for stars with highly non-circular
orbits.]  The vertical and radial actions may be coupled
(and thus the validity of the capture criterion is untested) for a
star that has large random motions.  Further, the
azimuthal force from the spiral (responsible for the changes in orbital angular momentum of a trapped star) will be weaker above/below the mid-plane. Our 
neglect of  vertical motions thus overestimates how strongly stars
in 3D orbits would interact with the spiral pattern.  We therefore expect
there is a systematic overestimate of the initial fraction of stars in
trapped orbits with increasing velocity dispersion.  As mentioned in
<ref>, <cit.> found an exponential decrease
in the RMS change in orbital angular momentum for an ensemble of stars
due to a transient spiral with increasing scale height of the
population.

We defer a more thorough discussion of the time-dependence of the fraction of stars captured in trapped orbits to a later paper in this series (Paper IV).






  


 §.§ Comparison to Previous Theoretical Work


A definition for maximum efficiency of radial migration from a single spiral pattern is the case in which  the RMS change in orbital angular momentum for the ensemble of stars in trapped orbits, ⟨(Δ L_z)^2⟩^1/2, is maximized over the spiral lifetime.  This definition depends on the fraction of stars in trapped orbits (evaluated using ℱ_Δ R in this paper), the amplitude of the change in each trapped star's angular momentum (maximum values can be estimated using the radial width of the capture region - R_max-R_min), and the degree of scattering out of trapped orbits on timescales less than the timescale for radial migration. 
  
<cit.> used a simulation to study the relationship between ⟨(Δ L_z)^2⟩^1/2 for a population of stars and that population's scale height or initial radial velocity dispersion.  They used seven tracer populations of test particles in a 3D disc with gravitational field defined by tapered, exponential thin and thick discs and two spiral arms.  In two separate experiments each population was assigned either a different scale height or initial radial velocity dispersion. 
They found that the value for ⟨(Δ L_z)^2⟩^1/2  for a given stellar population decreases exponentially with linearly increasing values for the scale height.  They also found that ⟨(Δ L_z)^2⟩^1/2 is smaller for populations with higher initial radial velocity dispersions.[<cit.> do not find a functional form for the relationship between the RMS change in orbital angular momentum (⟨(Δ L_z)^2⟩^1/2) and initial radial velocity dispersion (σ_r,0).  However, we find that the closest fit to the data goes as ⟨(Δ L_z)^2⟩^1/2∝σ_r,0^1/2.] 
It is encouraging that we also find a decline in the value for the integrated fraction of stars in trapped orbits, ℱ_Δ R, with higher values for the radial velocity dispersion (<ref>), strengthening the argument that the efficiency of radial migration depends on the radial velocity dispersion.

It is not obvious how to further compare our results to the trend
found by   The value of ⟨(Δ
L_z)^2⟩^1/2 is, in general, a time dependent quantity (for example, due to the effects of scattering) for the entire population of
stars.  <cit.> measured ⟨(Δ L_z)^2⟩^1/2
over the lifetime of the simulation in order to evaluate the final
⟨(Δ L_z)^2⟩^1/2 induced by a transient spiral
pattern from initial growth through final decay.  The integrated
fraction (ℱ_Δ R) we have derived in this paper is a measure only of the initial
fraction of stars that may migrate radially.  We therefore
interpret the fall off in the initial fraction of stars in trapped
orbits with increasing velocity dispersion as a first step toward
isolating the physics important to understanding the time-dependent
value of ⟨(Δ L_z)^2⟩^1/2.  The time dependence
for the fraction of stars in trapped orbits obviously depends on the rate of
scattering. Nonetheless, it is expected that the
effects of scattering will be less important for more open spiral
arms (since even a star with large radial excursions from its guiding
centre will not interact with the spiral away from corotation) and for
perturbations with fewer spiral arms (m=2 rather than m=4 – since the Lindblad
resonance will be farther from the radius of
corotation).[Figure 7 and Table 3 from <cit.> show that
the value for ⟨(Δ L_z)^2⟩^1/2 is smaller for
higher number of spiral arms, even when the prescription for the
initial radial velocity dispersion is smaller for higher number of
spiral arms (Table 1).]

Further, we cannot make any predictions about whether or not the vertical position or motion of a star affects its trapped status because we have assumed a 2D disc on the premise that vertical action and radial action are separable (see Paper I, 2).  Studies of simulated disc galaxies support the notion that the vertical velocity dispersion of a population negatively affects the degree to which it migrates radially <cit.>.
However, should the ratio σ_R/σ_z be approximately constant, we expect a similar trend in the initial fraction of stars in trapped orbits as in equation <ref>.

<cit.> predict that the maximum change in
angular momentum for a star that migrates radially, Δ L_max,
in a disc with a flat rotation curve is proportional to R 
|Φ_s|^1/2_CR.  <cit.> predict that the maximum possible
change in orbital angular momentum for any individual star in a
trapped orbit (Δ L_max) is set by the width of the
capture region <cit.>.
Figure <ref> illustrates the dependence of the maximum width of the capture region (R_max-R_min) on the fractional surface density for the spiral pattern (ϵ_Σ) and its radius of corotation (R_CR). We expect that the maximum change in angular momentum per unit mass will be  Δ L_max≈ v_c (R_max-R_min).
Whatever the functional form, the maximum possible change in angular momentum for a single star is of interest only in the sense that it is a limiting case, whereas the maximum amplitude for ⟨(Δ L_z)^2⟩^1/2 depends on the distribution of all the individual angular momenta of stars in trapped orbits.  Indeed, <cit.> find a positive correlation between the amplitude of a perturbation and the degree of induced radial migration in a suite of simulated disc galaxies.

<cit.> predict that radial migration will be more
important, for a given radial velocity dispersion σ_R, when the
value for Q (equation <ref>) at corotation approaches unity
since self-gravity strongly enhances perturbations to the
potential at corotation.  They develop a formalism which they use to
compute a diffusion tensor in a razor thin disc and find that
diffusion is parallel to the angular momentum axis at corotation,
consistent with radial migration.  In the series of models we use for
Figure <ref> we hold the radial
velocity dispersion profile constant and vary the radius of
corotation.  The radial dependence for Q(R) would then be given by 
Q ∝ (R
 e^-R/R_d)^-1 and would have a minimum value of Q∼
0.7 at R=R_d.  Our models do not sample values of corotation
inside R=4 kpc, but measures of the integrated fraction with
unconstrained radial range of evaluation increase steadily toward the
galactic centre.  In
Figure <ref>, Q = 1 at R =
5.5 kpc, corresponding to a more steep decrease in the values
for the integrated fraction at R≳ 5.5 kpc.  The
comparison between 's results and ours is not
direct.  argue that a lower velocity dispersion
leads to a stronger perturbation which stimulates more migration and
we argue that a stronger perturbation traps more orbits, but the
integrated fraction is lower for higher velocity dispersion.
Additionally, our Model Q, which specifically sets the radial velocity
dispersion so that Q=1.5 at all radii, has the lowest values
for radial velocity dispersion profile at all radii (illustrated in
Figure <ref>) and the highest values for the
integrated fraction at all corotation radii.  This is a somewhat
obvious result and would presumably be more evident for lower
values for Q.


Spiral patterns that corotate or nearly corotate with the disc can induce radial migration over large radial distances <cit.>.  
In <cit.> 4, we discuss how the radial range of the capture region depends on the radial rate of divergence between the spiral patten speed and the circular orbital frequency for stars; this divergence rate equals zero for corotating spiral patterns and thus the capture region spans the full radial range of the spiral arm.
The underlying assumptions for our models in this study have a divergence rate set by a flat rotation curve and radially constant pattern speed.  We have not explored other divergence rates, but we expect the integrated fraction to decrease with increasing kinematic temperature in all cases since the underlying physics will be the same no matter the size of the capture region.
A complication we have not addressed in this work is the consequence of overlapping resonances from multiple patterns <cit.> or multiple transient modes <cit.>.




§ CONCLUSIONS


This is the second of a series of papers seeking to quantify the
physical conditions under which  radial
migration is an important process in the evolution of disc galaxies.  The efficiency of radial
migration is directly related to the fraction of disc stars
captured in trapped orbits, which is investigated here.  We here apply the capture criterion we obtained in an earlier paper (a brief review is given in
Appendix <ref> of this work) to a series of models
of a spiral galaxy (described in <ref>) in order
to investigate how the adopted models and parameter values
affect the initial distribution of the fraction of stars in trapped
orbits (ℱ_R(R)) and the integrated fraction
(ℱ_Δ R) over one of several specified radial ranges
(Δ R).

For any given spiral pattern, ℱ_R(R) depends on the
distribution of orbital angular momentum near corotation
(R_CR) since whether or not a star is in a trapped
orbit is well approximated by whether or not its guiding centre
radius is within the capture region.  At any given radial coordinate
R, the velocity dispersion characterises the width of the azimuthal
velocity distribution, and thus the distribution of orbital angular
momentum at that radius.  The capture criterion specifies
the range of orbital angular momenta at a given spatial
coordinate for a star to be in a trapped orbit.  Therefore the
fraction of a population that meets the capture criterion at
radius R depends on the local velocity dispersion.  We use the
radial velocity dispersion as a proxy for the distribution of angular
momentum since it is more easily observed and quantified.  The
profile for ℱ_R(R) for populations with high velocity
dispersion has a radial range that is greater than the size of the
capture region since the radial epicyclic excursions (parameterised by the maximum amplitude, X) of these stars from
their guiding centre radii (R_L, related to orbital angular momentum by
equation <ref>) scales with the radial velocity dispersion (by
equation <ref>).  The peak of the distribution of the
fraction of stars in trapped orbits has a smaller amplitude for
populations with larger velocity dispersion.  The offset of the
peak of the distribution of stars in trapped orbits from the radius of
corotation (R_peak-R_CR) to larger radii is greater for
populations with higher velocity dispersion
(equation <ref>) due to the slower mean orbital
velocity associated with asymmetric drift
(<ref>).

The integrated fraction (ℱ_Δ R) is the
measure we use to quantify how the fraction of stars in trapped
obits depends on the radial velocity dispersion.  We evaluate
ℱ_Δ R by integrating the distribution of the trapped fraction over several specified  radial ranges (Δ
R), where we use two classification schemes for Δ R.  These
are the radial ranges that have either fixed radial width centred
around corotation (Δ R_1 and Δ R_2) or an unconstrained
radial range (Δ R_5%, Δ R_25%, and Δ
R_FWHM) defined by the range within which the value for
ℱ_R is greater than 5%, 25%, and half the
maximum amplitude of the fraction of stars in trapped orbits as a function of radius.
We use the combination of these fixed and unconstrained measures
for Δ R to gain the following insights into how the
distribution of stars in trapped orbits depends on the adopted model
and parameter values.

The integrated fraction is smaller for stellar
populations with higher velocity dispersion, consistent with
the trend found by <cit.> (see <ref>).
In Paper I, we analytically demonstrated that radial action, correlated with random orbital energy (i.e., the energy associated with non-circular motion), is mostly irrelevant to trapping.  Indeed, it is the azimuthal action, or angular momentum in the epicyclic limit, that is primarily responsible for trapping.  The kinematics of a population of stars at 2D coordinate 𝐱 can be described by the shape of a velocity ellipsoid having ratio σ_R/σ_ϕ.  The spread in the orbital angular momentum distribution (related to σ_ϕ at R through the individual angular momenta given by L_z = R v_ϕ per unit mass) is thus correlated with the radial velocity dispersion.  In Figure <ref>, we illustrate how the spread in orbital angular momentum and asymmetric drift determine the fraction of stars in trapped orbits at coordinate 𝐱.
Figure <ref> shows a linear fit
(equation <ref>) for the integrated
fraction (total fraction of stars in trapped orbits) as a function of radial velocity dispersion for each measure
of Δ R for the model that assumes constant σ_R.
We argue that a similar trend exists for any reasonable 
distribution function since the decrease in the integrated fraction fundamentally
stems from the broadening of the distribution of angular momentum.

The
fitting constants are given in Table <ref>, where the
maximum fraction of stars in trapped orbits over the radial range of
the capture region (ℱ_5% as σ_R→ 0) is
well approximated by ℱ_Δ 5%, max=2/π∼64%
(<ref>).  Further, we investigated the
effect of using a radially dependent radial velocity dispersion
profile (<ref>) and found that there
are some differences in the distribution of stars in trapped orbits,
but that the differences in the integrated fraction are not
particularly significant as long as the radial velocity dispersion
profile has a shallow radial gradient over the radial range of interest -  as is the case
for all models in this paper.

The distribution of stars (initially) in trapped orbits
for populations with high velocity dispersion has a large radial
range, spanning of order a few disc scale lengths.  This
is a consequence of the large radial excursions from the guiding
centre radius for stars that meet the capture criterion (that the guiding centre radius lie 
within the capture region).  However, the fraction of stars that meet
the capture criterion at large radial distances from R_CR is
small.

The maximum value for the integrated fraction ℱ_Δ R over a given radial range (Δ R) is determined by the amplitude of the spiral perturbation at corotation.

The integrated fraction (ℱ_Δ R) for the fiducial Models Σ, Q, and W,  is nearly independent (±∼ 10%) of corotation radius (R_CR) or patten speed (Ω_p = v_c/R_CR).  Each model assumes an appropriate radially dependent radial velocity dispersion profile (σ_R(R)).  The mean value for ℱ_Δ R in these models depends on the normalisation of the radial velocity dispersion profile and the assumed amplitude for the 
spiral pattern.

The RMS change in orbital angular momentum, ⟨ (Δ
L_z)^2⟩^1/2, around the radius of corotation of a spiral
pattern depends on the fraction of stars in trapped orbits, which we
have explored in this paper.  ⟨ (Δ L_z)^2⟩^1/2
also depends on the amplitude of the oscillations in orbital angular
momentum for stars in trapped orbits (where the maximum amplitude
oscillation for a guiding centre radius is given by the size of the
capture region) and the time-scale for maximising ⟨ (Δ
L_z)^2⟩^1/2 due the these oscillations.  Should the spiral
disrupt on the time-scale for which each of these ingredients is
maximal, ⟨ (Δ L_z)^2⟩^1/2 would be maximised and
radial migration would be most efficient for a single spiral.
In <ref>, we argue that time dependent processes are
an important part of understanding the efficiency of radial
migration from a single spiral pattern, and thus ⟨
(Δ L_z)^2⟩^1/2, and the initial fraction of
stars in trapped orbits should be treated as an upper limit.  We defer 
exploration of the importance of scattering processes, the
distribution of orbital angular momentum oscillation
amplitudes, the distribution of time-scales for trapped
orbits, and the consequences of a time-dependent spiral amplitude to 
later papers in this series.



§ ACKNOWLEDGMENTS

It is a pleasure to thank J. Sellwood, R. Schönrich and P. McMillan for helpful discussions during the course of this study.
We also thank the anonymous referee for thoughtful comments that have improved the quality of this work.
This material is based upon work supported by the US National Science Foundation Graduate Research Fellowship under Grant No. DGE-1232825, National Science Foundation Grants AST-0908326 and OIA-1124403, and an American Association of University Women Dissertation Fellowship.  RFGW thanks the Leverhulme  Trust for a Visiting Professorship, held at the University of Edinburgh. She is also grateful for support from her sister, Katherine Barber. This work was performed in part at Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611. RFGW thanks all at ACP for the wonderful environment for physics they have created and maintained. 


mn2e 




§ THE CAPTURE CRITERION


In Paper I, we derived a capture criterion" that can be used to
determine whether or not a star in an infinitely thin (2D) disc is in a trapped orbit.  As
defined in <ref>, trapped orbits occur near the
radius of corotation with a spiral pattern (or any density
perturbation) and are characterised by having oscillatory orbital
angular momentum with little to no change in random orbital energy (as also discussed in <cit.>).  A
star that is in a trapped orbit could migrate radially should the
spiral pattern be transient.  In this section we briefly review the
capture criterion, but refer the reader to Paper I for a more
complete derivation and discussion.

The capture criterion can be applied to any disc star given its 4D phase space coordinate (R,ϕ,v_R,v_ϕ), the amplitude of the spiral perturbation to the potential at the radius of corotation, |Φ_s|_CR, and the slope of the rotation curve at corotation <cit.>.

It is both informative and convenient to express the capture criterion in terms of orbital energy and orbital angular momentum.  The orbital energy in the inertial frame (E) and orbital angular momentum (L_z) can be combined to form an expression for a conserved quantity known as the Jacobi integral <cit.>,

    E_J = E - Ω_p L_z = 1/2|𝐱̇|^2 + Φ_eff(𝐱) ,

with

    Φ_eff(𝐱) ≡Φ(𝐱) -1/2|Ω_𝐩×𝐱|^2

being the effective potential <cit.>, and 𝐱 and ẋ being, respectively, the position and velocity of the star in the rotating frame.  Since the spiral potential is time-independent in the rotating frame, it is in this frame that the Jacobi integral is conserved.  

Figure <ref> shows a Lindblad diagram for stars in a disc with a flat rotation curve.  The Lindblad diagram illustrates the relationship between the Jacobi integral and the conservation of orbital circularity at corotation.  The time derivative of equation <ref> shows that changes in a star's orbital angular momentum are linearly related to changes in its orbital energy with slope equal to the spiral pattern speed (represented by a dashed line in Figure <ref>).  In a disc with a flat rotation curve, stars in circular orbits (solid curve) have logarithmically increasing orbital energy with linearly increasing orbital angular momentum, where vertical distance from this curve represents random orbital energy.  At corotation (shown as an X' in Figure <ref>), the slope of the line representing pattern speed is parallel to slope of constant orbital circularity.  Since the Jacobi integral (E_J) is conserved along lines with slope equal to the pattern speed (Ω_p), stars in trapped orbits do not have significant changes in orbital eccentricity with changes in orbital angular momentum around corotation.




The generalized capture criterion is derived using action variables, but in order to write an expression in terms of orbital energy and angular momentum we assume the epicyclic approximation.  This approximation describes mildly non-circular orbits as simple harmonic oscillations about a circularly orbiting guiding centre.

The epicyclic frequency of radial oscillations is given by <cit.>,

    κ^2(R_L)=( RdΩ_c^2dR+4Ω_c^2)_R_L ,

where Ω_c is the circular orbital frequency, and the mean orbital radius[Note that this approximation is only strictly true for SHM, requiring radial oscillations be within a range where the surface density is approximately constant.] (i.e. guiding centre radius") is

    R_L = R v_ϕv_c.

The random orbital energy in the inertial frame is

    E_ran=E-E_c(R_L),

where E_c(R) is the energy associated with a circular orbit in the underlying axisymmetric potential at radius R.

The form of the capture criterion appropriate for our assumed model is <cit.>

    -1 < Λ_nc,2(t) ≤ 1 ,

where <cit.>

    Λ_nc,2(t) ≡E_J(R,ϕ,v_R,v_ϕ)-h_CR|Φ_s|_CR - (R_L(R,v_ϕ,t)R_CR) ( E_ran(R,ϕ,v_R,v_ϕ,t)|Φ_s|_CR)

and h_CR is the Jacobi integral (equation <ref>) for a star at corotation in the underlying axisymmetric potential (Φ_0(R_CR)).



