CossamDoppler : simple examples


green Example 1 : "Virtual and spurious surface structure on Ap stars"
                        by M.J. Stift, in IAU Symp. No. 176   Stellar Surface Structure
                        eds. K.G. Strassmeier and J.L. Linsky, Kluwer, Dordrecht, p. 61-68


More than 2 decades ago, Figs. 2 and 3 of Stift's article demonstrated what happens when a very moderate magnetic field is neglected or when an incorrect magnetic field geometry is adopted in Zeeman Doppler mapping.

These results have consistently been ignored by the entire ZDM community up to the present day.

Back in 1996, the results were obtained with a code written by R.M. Fensl in Ada83. In the meantime, an independently developed new ZDM code based on Ada95 and Ada2012 takes advantage of the huge increase in computing resources over the last 20 years. Wouldn't it be instructive to see how today's results compare to those published in 1996?

For simplicity's sake we take an atomic transition that splits into a simple Zeeman triplet and a stellar atmosphere with Teff = 9000K and log g = 4.00. The magnetic geometry adopted for synthesising the phase-dependent Stokes profiles is the same as in Stift (1996), i.e. an oblique rotator with {i, α, β, γ, x2, x3} = {70.0, -90.0, 0.0, 0.0, 0.20, 0.00}. The profiles (and related files) calculated with the help of CossamMulti

    ./cossam_multi  cossam_multi_01.inp  7020650790  7020650790

can be found here.

We first look at the example shown in the left part of Fig.2 in Stift (1996), then at the right part of Fig.2 and finally at the right part of Fig.3.

        /cossam_doppler  cossam_doppler_01.inp  7020650790  7020650871  >  codop_01.res &

        /cossam_doppler  cossam_doppler_02.inp  7020650790  7020650871  >  codop_02.res &

        /cossam_doppler  cossam_doppler_03.inp  7020650790  7020650871  >  codop_03.res &

The results are stored in the following three sub-directories

       702065155_dir

       702065198_dir

       702065248_dir

You may use auxiliary programs provided with CossamDoppler to visualise the results. Invoking

       ./hammer_nogrid_inv_vs_ori_diff   702065155_dir/dopimag_ham.702065155K

       ./hammer_nogrid_inv_vs_ori_diff   702065198_dir/dopimag_ham.702065198K

       ./hammer_nogrid_inv_vs_ori_diff   702065248_dir/dopimag_ham.702065248K

provides you with plots of the difference  inverted abundance map minus original abundance map (in this case perfectly homogeneous over the whole stellar surface).

A complete set of files can be downloaded here.

 

green Example 2 : A simple abundance ring following the magnetic equator of a centred dipole
                       inclined by 90 degrees with respect to the rotational axis (oblique rotator)


Convergence is not always as easy to achieve as in Example 1. Sometimes the problem is quite ill-behaved, convergence is slow, and the right choice of parameters (weight, step-size, starting abundance) is not straightforward; it may involve attempts with dozens of different combinations before you get it right.

Warped rings resulting from non-axisymmetric magnetic geometries are notoriously difficult to recover (see Stift & Leone 2017), but in this simple case of an abundance ring following a meridian of the magnetic star you can verify yourself that CossamDoppler achieves results perfectly comparable to what the INVERS family of codes is able to deliver.

For this purpose there is no need to use 7 lines at 20 equidistant phases, nor do you have to push the spectral resolution to 0.02 Angstroms. Just try this one (2 lines, 10 phases, 0.05 A) and compare the abundance errors  CossamDoppler vs. INVERS.

    ./cossam_doppler  cossam_doppler_10.inp  7012365600  7021870909                                        > codop_10.res &

    ./cossam_doppler  cossam_doppler_11.inp  7012365600  7021870909  702187240_dir/dopimag_ham.702187240J  > codop_11.res &

    ./cossam_doppler  cossam_doppler_12.inp  7012365600  7021870909  702187291_dir/dopimag_ham.702187291G  > codop_12.res &


Let me explain. For each run, the rms scatter reaches a minimum at some iteration stage. If this rms scatter is still larger than what you expected or desired, restart CossamDoppler with the dopimag_ham file corresponding to this best map as the 4th argument. The program control (input) file has to be modified accordingly, usually adopting smaller weight and step size. In the present example, the 3rd run yields a final best fit to the "observed" line profiles of better than 0.1%.

Note: Counting of the individual dopimag_ham files starts in row 8 of dopimag_res.
          The contents of  702187240_dir/dopimag_res.7021872408  (with the last character of dopimag_ham added for clarity):
Number of grid points     :   2034
Sum(epsilon)              :    2.372E+11
Entropy                   :    7.541E+00
Minimum field strength    :     1250.0
Maximum field strength    :     2497.1
Maximum absolute velocity :     39.94
Epsilon_total  Entropy_total  Chi_square_total :        2.3718E+11      7.541152E+00      3.580905E-02      7.541152E-04 
Epsilon_total  Entropy_total  Chi_square_total :        3.2513E+11      7.503768E+00      1.013834E-02      5.368913E+00  A
Epsilon_total  Entropy_total  Chi_square_total :        3.0108E+11      7.473336E+00      5.021450E-03      6.415957E+00  B
Epsilon_total  Entropy_total  Chi_square_total :        3.2660E+11      7.437508E+00      3.361881E-03      6.729589E+00  C
Epsilon_total  Entropy_total  Chi_square_total :        3.1878E+11      7.427129E+00      2.939036E-03      6.808249E+00  D
Epsilon_total  Entropy_total  Chi_square_total :        3.2758E+11      7.412979E+00      2.594284E-03      6.866694E+00  E
Epsilon_total  Entropy_total  Chi_square_total :        3.2497E+11      7.408771E+00      2.459631E-03      6.890841E+00  F
Epsilon_total  Entropy_total  Chi_square_total :        3.2858E+11      7.402009E+00      2.302133E-03      6.917243E+00  G
Epsilon_total  Entropy_total  Chi_square_total :        3.2804E+11      7.398922E+00      2.216519E-03      6.932184E+00  H
Epsilon_total  Entropy_total  Chi_square_total :        3.2983E+11      7.394817E+00      2.124054E-03      6.947550E+00  I
Epsilon_total  Entropy_total  Chi_square_total :        3.3598E+11      7.370432E+00      1.683324E-03      7.015970E+00  J
Epsilon_total  Entropy_total  Chi_square_total :        3.3572E+11      7.371559E+00      1.692692E-03      7.015124E+00  K
A complete set of files can be downloaded here.


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