departure coefficients in cooling sims

Marios Chatzikos

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Mar 29, 2021, 12:59:38 AM3/29/21

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Hello,

Back to the time-dependent cooling sims.

Since we are considering using state-resolved advection in place of
ionization
balance advection, it would be good to see how other aspects of the
physics are
affected by this change.

I have been looking at how departure coefficients change with temperature.
Naively, I would expect them to be significantly different from those
obtained
with ionization-balance advection, and so, in principle, from the results of
Storey & Hummer (1995):

https://ui.adsabs.harvard.edu/abs/1995MNRAS.272...41S

SH95 present results for varying electron densities and temperatures for all
elements up to oxygen. What is of interest to me in these calculations are
the l-resolved departure coefficients for n<=10, b_nl, and the collapsed
coefficients, b_n, for all levels.

To compare with SH95, I ran a number of cooling simulations with the
hydrogen
density ranging from 2 to 13. I then extracted the departure
coefficients for
timesteps that were close (better than 1 per cent) to the SH95
tabulations for
the temperature, and reasonably close (better than 50 per cent) for the
electron
density. Because of these requirements, comparisons are sparse.

The attached file shows the results. It is almost 200 pages long, each page
presenting the b_nl and b_n distributions computed with state-resolved
advection
with the default number of levels (labeled 'state-res'), with 90 collapsed
levels ('state-res-nc90'), and ionization advection ('ion-bal'), against
those
of SH95.

In terms of the state-resolved advection results, mixed conclusions may
be drawn.

For H, at low temperatures (500 & 1000K), the agreement of b_nl and b_n is
decent for low densities (log ne < 5), but breaks at higher values. At
higher
temperatures, the situation reverses, and for most densities, the b_nl
are in
good agreement, particularly for n<=5 or so; however, the agreement for
higher
levels is not good at low densities, but improves at higher densities.

Similar remarks apply to He.

For higher Z, timesteps of similar conditions to SH95 were found only for
temperatures of 30, 50, and 100 thousand K. The b_nl do not show the
variation
with n,l that the SH95 results have (and the Cloudy results for H, He
also had),
but are generally in agreement for n~<5.

Increasing the number of collapsed levels to 90 (for a total of 100)
does remove
some unphysical tails (n > 15-20) from the above results, and recovers
departure
coefficients of around 1 at the limit of large n.

Ionization-balance advection produces results nearly identical to
state-resolved
advection, except for temperatures of 1e4 K, and log ne < 7, where it
produces
results closer to SH95 for n <= 10, but more divergent for n > 15. At
higher
temperatures and low densities, it leads to deviations by factors of 10-100.

It is surprising these results are as good as they are. I remember the
steady-
state departure coefficients were not in good agreement with SH95. I
included
steady-state results in these plots for comparison, and they confirmed my
recollection. I'm not sure if we'd looked at the departure coefficients
over
as wide a parameter space of plasma conditions before -- we focused on
10,000 K
when we were working out bugs a few years back, if I recall correctly.

I would like to discuss the departure coefficients in the paper I'm
writing, but I'm
not sure how much I should believe them. The fact that the steady-state
results
are all over the place is not encouraging to begin with, but the problem
there
is the lack of state-specific recombination. The dynamical rates dominate
recombination by a factor of ~250 for (log ne, T)=(3, 500), Fig 1. That
should
probably give us some confidence that the results are rather robust,
i.e., had
state-selective recombination been used, we would have recovered pretty
much the same results.

Am I reading this right? What are your thoughts?

Thanks,
Marios

comp-bns.pdf

Marios Chatzikos

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Apr 8, 2021, 3:57:19 PM4/8/21

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I didn't get much of a response on this. It was probably too long to
read an e-mail.

The gist is that the departure coefficients in cooling sims with
state-specific advection are pretty close to the SH95 results, which
seems suspicious. The comparison against the steady-state results was
intended to illustrate the effect the dynamical terms have.

The comparison is meaningless. The steady-state results are nowhere
near the SH95 results -- see attached (page 6 holds the H results at
10,000K).

The solver seems to be in a dubious state.

Marios

T_plots_steady.pdf