Mathematical Reconstruction of Axial Star Morphology & Kinematics in Space

1. Space reconstruction of the morphology and kinematics of axisymmetric radio sourcesP.N. Diep*,N.T. Phuong,D.T. Hoai, P.T. Nhung, N.T. Thao, P.Tuan-Anh and P. DarriulatDepartment of Astrophysics (DAP)Vietnam National Satellite Center (VNSC)Quy Nhon, August 2016

2. THE PROBLEM Aim: address mathematical issues related to the reconstruction in space (deprojection) of the CSEs of axial stars Restriction: evolved stars and protostars in mm/sub-mm (typically CO) Input data are sky coordinates, y and z, Doppler velocity along the line of sight x, Vxand flux density f(y,z,Vx). Quantities to evaluate are space coordinates, x, y and z, density d, temperature T, velocity components Vx, Vyand Vz. Eight instead of four: we need additional assumptions.

3. x ξ (Star axis) θ y’ ψ η y (East) θ ζ z’ COORDINATES Line of sight: x Sky plane: East y, Northz Polar coordinate of pixel on the sky (R,φ) R=(y2+z2)½ ; tanφ=z/y Star axis: ξ makes an angle θ with x projects on sky plane at angle ψ with z Stellar latitude α Stellar longitude ω ξ=xcosθ+Rsinθsin(φ–ψ) η=Rcos(φ–ψ) ζ=–xsinθ+Rcosθsin(φ–ψ) z (North) r=(x2+y2+z2)½

4. For most of their life, AGB stars have a small mass-loss rate, ~10‒7 solar masses per year, the shape of the gas volume is not much affected, the bipolar outflow is seen mainly on the wind velocity The velocity components are a radial component Vrad an axial component Vax a rotation componentVrot Under the hypothesis of rotation invariance about the star axis, theyare functions of r and sinα exclusively, independent of ω Vx=xVrad/r+cosθVax−Rcos(φ–ψ)sinθVrot/(rcosα)

5. We introduce an effective emissivity ρ ρ(x,y,z)=f(y,z,Vx)dVx/dx, implying ∫f(y,z,Vx)dVx=∫ρ(x,y,z)dx=F(y,z) ρ is a function of: the gas density, the abundance of the emitting gas, the population of the emitting quantum state and the probabilities of emission and absorption of the detected radiation. It replaces d and T. It saves another degree of freedom in addition to rotational invariance. T and d can be evaluated separately if we have data on two molecular lines.

6. Case A Case B Vrot Vrad α α (star latitude) 0 45o 90o 0 45o 90o (Equator) (Pole) Simulations are for a centrally symmetric star, ρ(x,y,z)=ρ(–x,–y,–z)V(x,y,z)=–V(–x,–y,–z)with rotation symmetry about the star axisand no axial wind (radial expansion+rotation)ρis taken proportional to r–2For each of two cases (A & B), study is made for: pure expansion, expansion+rotation and pure rotation V0

7. An example: P-V diagrams, φ vs Vxfor Case A, θ=75o, ψ=0 Pure expansion Mixed rot/exp Pure rotation 2π π 0 π φ −10 0 10 −10 0 10 Two signs of V at any given φ V (km/s) Complex combination of expansion and rotation Central symmetry: Changing φ in φ+π leaves ρ invariant and changes the sign of V Only one sign of V at any given φ P.N. Diep et al., MNRAS 461, 4276 (2016)

8. Mixing rotation and expansion is simple incase B: x probes either inside the cone where it sees only expansion or outside where it sees only rotation. 2π π 0 φ θ=45o −10 0 10 −10 0 10 Pure expansion Mixed rot/exp Pure rotation Mixed configurations: simply the superposition of pure expansion and rotation −10 0 10 V (km/s) 8

9. The sky map of <Vx> is not sufficient to evaluate ψSpiraling mixes the effects of rotation and expansionTo tell rotation around axis A from expansion along axis B perpendicular to A we need a more refined method.

10. Evaluating the value of the position angle ψ

11. Inclination θ of the star axis with respect to the line of sight with Modeling Writing

12. Telling rotation from expansion sum extends over ±Vx Pure rotation Pure expansion Average over R: Pure expansion Pure rotation Aas(φ)=0 for both θ = 90◦ and θ = 0◦ and is always smaller than unity. Independently from both ϕ and θ The value of A(ϕ) at ϕ = 0◦, ϕ = 90◦ or its integral over ϕ should be good indicators of the relative importance of rotation over expansion

13. Radial dependence of the CSE properties • ρ(x,y,z)~r−nproduces a ∫ρ(x,y,z)dx ~R−(n−1) on the sky map • These results illustrate the advantage of displaying the measured flux densities in terms of polar rather than Cartesian coordinates on the sky plane: • Their dependence on R is related simply to the dependence on r of the effective emissivity and wind velocity, • Their dependence on ϕ is related to the dependence of the effective emissivity and wind velocity on ψ, θ and αwithout having recourse to a model • Much information about the morphology and kinematics of the CSE can be obtained from simple quantities constructed directly from the expression of the flux densities in polar coordinates.

14. Application to a real AGB star: RS Cnc seen in CO(1-0)(Plateau de Bure observations) D.T. Hoai et al., A&A 565 (2014) A54, P.T. Nhung et al., RAA 15 (2015) 713 fit data 0.03 5 0 −5 0.1 RF(y,z) z (arcsec) RF(y,z) 0.01 0 0 180 360 φ (o) 0 2 4 6 8 R (arcsec) 5 0 −5 y (arcsec) Sky map of RF(y,z) No visible elongation about star axis R-dependence of RF(y,z) averaged over φ the fit is a power law of index −0.63 ( ~ −1.63 in r) φ-dependence of RF(y,z) averaged over R the fit is a sine wave of amplitude 4.0 10−4, (1.3% of the mean value)

15. Sky map of the mean Doppler velocity <Vx>showing a clear north-south asymmetry, evidence for the bipolar outflow φ-dependence of <Vx> averaged over R; the fit is a sine wave of amplitude 0.92 kms−1 and phase of 6o R-dependence of <|Vx|> averaged over φ; the fit is linear, with a slope of −0.12km/s per arcsecond 1 0 −1 4 3 2 fit data 5 0 −5 <|Vx|> (km/s) <Vx> (km/s) z (arcsec) 0 3 6 R (arcsec) 0 180 360 φ (o) 5 0 −5 y (arcsec)

16. 360 180 0 10 5 0 fit line φ(o) χ2 −10 0 10 Vx(km/s) −450 45 ψ (o) Position-velocity diagram, φvs Vx Expansion is seen to dominate over rotation Dependence on ψof a quantity χ2that has been constructed to measure the value of ψ; Much useful information can be obtained from proper inspection of the data without having recourse to a model

17. The Red Rectangle [CO(6-5), CO(3-2)] P.Tuan-Anh et al., RAA 15, 2213 (2015) A protostar: L1527 [C18O(2-1)] P.Tuan-Anh et al., submitted to MNRAS (2016)

18. Conclusions • Before undertaking modelling, a firstclose look at the data without requiring the help of a model, inparticular, the sky maps of RF(y, z), of Vx and of |Vx| and theirprojections on the R and ϕ axes provides essential information. • Use should be made ofsimple quantities that carry important information. Examples havebeen given, such as χ2ψ and A(ϕ), which may help the evaluationsof the orientation of the star axis in space and of the relative contributionsof expansion along the star axis and rotation about it. • We have argued thatusing P–V diagrams in polar rather than Cartesian coordinates is oftenbetter suited to making the best of them. 18

19. Thank you for your attention! Cảm ơn nhiều!