Background and Set-Up
Eilers and Hogg have created "global" stellar kinematic map of the Galactic disk, with 6D information on ~20.000 stars extending in RGC from 0 to 20 kpc. Between 6kpc and 13 kpc [TBC]
this map shows nice "spiraly" radial velocity pattern, that we should attribute to a "spiral perturbation".
The question is what practical ways are to interpret these data in terms of (dynamical) spiral arm perturbation (its strength, pattern speed and morphology).
These notes hare have emerged from conversations between C. Eilers, J-B Fouvry and H-W Rix in HD, and serve (among other things) to bring D. Hogg in the loop.
The observational information
This "map" is actually n(R,phi,z,vphi,vR,vz) == f(R,phi,z,vphi,vR,vz) x S(R,phi,z,vphi,vR,vz),
where S(R,phi,z,vphi,vR,vz) is the selection function.
The main issue with the selection function is that it separates into
S(R,phi,z,vphi,vR,vz)=S(R,phi,z) x S(vphi,vR,vz), where S(R,phi,z) = very complex, and S(vphi,vR,vz)=const. The selection function can be writen as S (D | phi,theta)*S(phi)*S(theta), where
R=Ro-D*cos(l) cos(theta) and z=D*cos(theta); S(phi) and S(theta) are highly structured, S(D | ..) is (kind-of) smooth.
Rather than modelling the fill distribution function in action-angle space F(J,theta) [the "other theta"],
we se whether there is a sub-space, where the selection function is flat, taking advantage of S(vphi,vR,vz | D,phi,theta)=const.
The proposal here is to model the radial actions and radial angles; as their experimental selection effects may be benign.
Conceptual modelling approach:
We observed a distribution of stars in the disk, which is approximately axisymmetric, but has significant non-axisymmetries (see above). We would like to link that to a driving non-axisymmetric, time-dependent perturbation, focussing on the region >5kpc, as the pattern speeds are slow enough
that co-rotation (and hopefully other resonances) is 'far out'. [JBF stresses the importance of this!]
The problem of linking a time-dependent, non-axisymmetric potential to the resulting non-axisymmetric distribution function has been essentially (theoretically) solved (in the linear, and non-self-consistent regime) by Monari & Famaey (2016), building on Section 5.1. in BT.
If one has solved the axisymmetric problem, Phi_0 and f_0, one can write for a small/linear deviation from it:
which then leads to the following select steps:
Propose a periodic (in phi) potential pewrturbation
then
spelled out in cylindrical coordinates
resulting in a (formally) closed solution with:
and other terms. Basically all this is spelled out in MF16.
Conceptually, the miracle is that the explicit time-dependence can be eliminated to a seeming steady-state solution in a co-rotating frame (Eq 15 --> 16 in MF16).
Implementation issues:
Here are some suggestions (to Christina) how to implement it:
-- go through the Monari&Famaey paper and re-write/simply it to the 2D case of a razor-thin disk;
probably a good idea to wrapy your mind around the math.
-- let's choose a very simple Phi_1 perturbation functional form, to have something specific.
-- we need to get Phi_0 and F_0!
let's presume that for the moment we only consider F(J_R,theta_R), i.e. the radial action/angle (for the reasons above).
How do we then get F0(J_R,theta_R) = F0(J_R)?
One way would be to determine Phi_0 and F0(J_R) simultaneously through 'orbital roulette' (== most plausible angle distribution); this may be an intereting project in itself:
Or, we just adopt Phi_0(R) from Eilers,Hogg,Rix,Ness2019, and fit for F0(J_R).
Note that we need to take subsequently derivatives of F0(J_R) w.r.t. J_R; this means we need
to take (and fit) an analytic form; presumably a pseudo-isothermal from Binney (see Trick, Bovy etc.. 2017).
Envisioned outcome:
-- decide whether there is a plausible geometry for Phi_1(R,phi,t) that matches the data above.
-- if so, ask what Phi_1 amplitude is implied and put it in astrophysical context (simulations etc..)
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