Multiwavelength campaign on NGC7469 II.
Column densities and variability in the Xray spectrum
We investigate the ionic column density variability of the ionized outflows associated with NGC 7469, to estimate their location and power. This could allow a better understanding of galactic feedback of AGNs to their host galaxies. Analysis of seven XMMNewton grating observations from 2015 is reported. We use an individualion spectral fitting approach, and compare different epochs to accurately determine variability on timescales of years, months, and days. We find no significant column density variability in a 10 year period implying that the outflow is far from the ionizing source. The implied lower bound on the ionization equilibrium time, 10 years, constrains the lower limit on the distance to be at least 12 pc, and up to 31 pc, much less but consistent with the 1 kpc wide starburst ring. The ionization distribution of column density is reconstructed from measured column densities, nicely matching results of two 2004 observations, with one large high ionization parameter () component at , and one at in cgs units. The strong dependence of the expression for kinetic power, , hampers tight constraints on the feedback mechanism of outflows with a large range in ionization parameter, which is often observed and indicates a nonconical outflow. The kinetic power of the outflow is estimated here to be within 0.4 and 60 % of the Eddington luminosity, depending on the ion used to estimate .
1 Introduction
Active galactic nuclei (AGN) are the most persistent luminous objects in the universe. Observed in all wavelengths from Radio to Xrays, they are powered by accretion of matter on to a super massive black hole.
Among the plethora of phenomenon they exhibit, 50% of type 1 AGN feature ionized outflows. The launching mechanism of these winds remains in debate, and suggestions vary from thermal evaporation (Krolik & Kriss, 2001) to line driving (Proga et al., 2000) and magnetic hydrodynamics (Fukumura et al., 2010). These AGN winds are observed in a multitude of absorption lines of different ions, in both UV and Xrays (Crenshaw et al., 2003). These lines are ubiquitously blueshifted with respect to the restframe of the host galaxy, with velocities often consistent between the Xrays and the UV (e.g. Gabel et al., 2003), suggesting they are part of the same kinematic structure.
If these outflows are indeed associated with the AGN, an important question is whether the energy or mass they deposit is important for galactic evolution by means of energy feedback. The kinetic power of these outflows scales with , the outflow velocity, which is typically a few 100 km s (Kaastra et al., 2002). These low velocities limit the efficiency of these outflows as a feedback mechanism. However, some outflows feature velocities of a few 1000 km s, NGC 7469 for example exhibits a fast component at a blueshift of 2000 km s.
AGN winds have been the focus of studies relating change in absorption troughs in AGN spectra to the distance and density of the associated outflows. Examples in both Xrays and UV analysis can be found in Behar et al. (2003); Gabel et al. (2005); Kaastra et al. (2012); Arav et al. (2015); Ebrero et al. (2016); Costantini et al. (2016). Arav et al. (2015) for example constrain the distance of the outflow in NGC 5548 to be at least a few pc from the AGN source, with distances up to more than 100 pc. These large distances lead to an ambiguity of whether the AGN is responsible for driving these outflows directly.
This is the second paper as part of a multiwavelength observation campaign on NGC 7469. Behar et al. (2017) derived outflow parameters using global fit models of photoionized plasmas. We continue the examination of the XMMNewton RGS spectrum focusing on measurement of the column densities. In addition to the seven observations observed on a logarithmic timescale during the 2015 campaign, we analyze archival data from 2004. With these data we compare changes on timescales of years, months and days, with the intent of seeking variability in absorption troughs, and through this to constrain the distance of the outflow from the AGN. This, along with a measurement of the kinetic power of the outflow will determine the role the outflow plays in coupling the AGN to its host galaxy.
2 Data
XMMNewton observed NGC 7469 as part of the multiwavelength campaign 7 times during 2015 for a total duration of 640 ks. The observation log is shown in Table 1, including previous observations published in Blustin et al. (2007). We use the RGS (1 and 2) data from all observations to constrain variability in absorption troughs. The RGS spectra are reduced using ‘rgsproc’ within the software package SAS 15^{1}^{1}1http://xmmtools.cosmos.esa.int and combined using the standard RGS command, ‘rgscombine’. The reduction is detailed in Behar et al. (2017). The spectral fitting in the present paper is done on grouped spectra, rebinned to 20 mÅ (grouping two default SAS bins). The full 2015 RGS spectrum (black) and bestfit model (red) are shown in Fig. 1, and the model is described in Sec. 3.1.
obs. Id  start date  RGS  exposure  

cts  ks  
a  0207090101  2004Nov30  1.42  84.7 
b  0207090201  2004Dec03  1.13  78.8 
1  0760350201  2015Jun12  1.36  89.5 
2  0760350301  2015Nov24  1.41  85.6 
3  0760350401  2015Dec15  1.18  84.0 
4  0760350501  2015Dec23  0.97  89.5 
5  0760350601  2015Dec24  1.04  91.5 
6  0760350701  2015Dec26  1.19  96.7 
7  0760350801  2015Dec28  1.23  100.2 
The EPICpn lightcurve of NGC 7469 is presented in Fig. 2. An interesting feature is the rapid change of photon flux on an hourly basis, while the average seems to remain constant over years.
The mean EPICpn count rate (count s) for the 2004 observations is 24.7, with a standard deviation of , and for 2015 the mean is 23.2 with .
3 Spectral modeling
3.1 Method
We first model the 2015 combined spectrum since column densities between observations in the campaign are consistent within 90% uncertainties (See Sec. 3.2). This agreement between the different observations, within the larger uncertainties of individual observations, is a clear indication in favor of using the combined spectrum, at least initially. All uncertainties we quote in this paper are 90% confidence intervals.
Following the ionbyion fitting approach by Holczer et al. (2007), we fit the continuum along with the ionic column densities, , which are this paper’s main goal. The transmission equation is given by:
(1) 
where is the observed continuum intensity, is the unabsorbed continuum intensity, is the absorption cross section depending on photon energy. The covering fraction is , with indicating no absorption and indicating the source is entirely covered by the outflow. Some results in the UV suggest the covering fraction is ion dependent or even velocity dependent (e.g Arav et al., 2012), but the much smaller Xray source is not expected to be partially covered. The Xray continuum of NGC 7469 in the RGS band can be modeled by a single powerlaw. A complete Xray continuum model based on the EPIC spectra will be presented by Middei et al. (in preparation). The powerlaw is given by:
(2) 
with the norm and the slope as free parameters.
On top of the absorbed continuum we observe emission lines. These lines were modeled by Behar et al. (2017), and include both photodriven and collisionally excited lines. They are fixed in our model and are assumed not to be absorbed by the outflow.
The absorption cross section is given by
(3)  
(4) 
Here describes the ionization edge of ion , is the Voigt line profile and the sum is over all the strong ion line transitions , is the electron charge, the electron mass, and are the oscillator strengths. All transitions are assumed to be from the ground level. We use the oscillator strengths and ionization edges calculated using the HULLAC atomic code (BarShalom et al., 2001) as used in Holczer et al. (2007).
The parameters determining the profile shape and position are ion temperature, turbulent velocity, and outflow velocity. The temperature and turbulent velocity broadenings seen in the UV (Scott et al., 2005) are below the RGS resolution of mÅ. Thus, in order to constrain simultaneously the covering factor, the turbulent velocity, and the ion column density one needs 3 measurable lines of a given ion (See Eq. 1). N is the best ion providing 3 lines unambiguously visible in the spectrum. These are observed at wavelengths of 25.18Å, 21.25Å, and 20.15Å. Nonetheless, the best fit favors a covering factor of 1.0 with the 90% confidence interval ranging down to 0.8 when all ions are taken into account. The uncertainty in the continuum adds another level of uncertainty here, so we make no claims regarding covering factor and hold it frozen to 1.0.
Since constraining the line profile parameters is not the goal of this paper, we fix the ion temperature at 0.1 keV. We then fit only the O Ly doublet line at the observed wavelength of 19.2Å with the outflow and turbulent velocities thawed and set initially to the values of Behar et al. (2017) in order to determine them. Fig. 3 shows the contribution of each velocity component to the absorption profile. The fit favors a 3 velocity model over 2 in accordance with these two papers, decreasing reduced (d.o.f.) by 0.5 from the 2component to the 3component model. The bestfit three components have velocities of 620, 960, and 2050 km s and turbulent velocities of 80, 40, 50 km s respectively. Three components are also favored by Scott et al. (2005) and Behar et al. (2017). Though the fit converges we are not able to obtain meaningful uncertainties on these parameters. We leave them frozen for the rest of the fit, freeing them for one final iteration after the ion column densities are constrained.
The fitted model parameters are thus the powerlaw normalization, the powerlaw slope, and the column density per ion. In addition, the three outflow velocities and three turbulent velocities are constrained once at the beginning according to O, and one more time at the end. The strength of this model^{2}^{2}2The code for the model can be found in https://github.com/uperetz/AstroTools, including a full graphical suite for fitting models to fits files. The README details the contents of the directory. lies in the independence of the ionic free parameters.
3.2 Column densities
The full 2015 spectrum (black) and bestfit model (red) is seen in Fig. 1, with a bestfit reduced of 1.4. For the 2004 spectra we obtain a reduced of . There are 1450 spectral bins and 64 free parameters. We also remeasure column densities from the 2004 spectra previously done by Blustin et al. (2007). This is done in order to maintain consistency in the comparison with the 2015 spectra using the same code and atomic data. Blustin et al. (2007) finds 2 velocities, but we retain the 3 velocity model for a consistent comparison with 2015. There is no increase of reduced compared to the two velocity model, suggesting the kinematics remain similar over a timescale of years. In Table 2 the continuum parameters of both epochs, 2004 and 2015, are presented.
2015  2004  
Flux  
ph keV s cm  
erg s cm, RGS band 0.31.5 keV (837Å) 
Finally the summed (across velocity components) column densities of the two epochs are given in Table 3. These are compared graphically in Fig. 4, as well as with the Blustin et al. (2007) measured column densities for reference. While the different velocities may be associated with different physical components, the current measurement is not sensitive to ionic column density changes in individual components due to the limited spectral resolution. This is manifested in an inherit degeneracy of column densities between the velocity components, and the sum allows us to increase the sensitivity to change.
A clear match can be seen, with 30/34 ion column densities within 90% confidence. Only N, O, Fe, and S are discrepant between observations, but with 90% uncertainties 34 measurements are expected to be discrepant. Moreover, other similarionization ions do not vary, indicating no absorption variability between the two epochs.
3.3 Absorption Measure Distribution
We characterize the ionization distribution of the absorber plasma using the Absorption Measure Distribution (AMD Holczer et al., 2007), defined as:
(5) 
where is the column density and is the ionization parameter. Here is the electron number density and is the distance of the absorber from the source. We can reconstruct the AMD using the measured ionic column densities:
(6) 
where is the solar abundance of the element (Asplund et al., 2009) and is the fractional abundance of the ion as a function of . We use a multiple thin shell model produced by XSTAR version 2.38^{3}^{3}3http://heasarc.gsfc.nasa.gov/docs/software/lheasoft/xstar/xstar.html, along with AMD analysis code in https://github.com/uperetz/AstroTools, see README. to determine the ionic fractions as a function of . The thin shell model assumes each is exposed to the unabsorbed continuum directly. This is justified by observing that the broad band continuum is not significantly attenuated by the absorption as seen by the relatively shallow edges (See Fig. 1). Our model grid is calculated from to with . We use a Spectral Energy Distribution (SED) from 1 to 1000 Ry extrapolated from our multiwavelength observations and corrected for galactic absorption (M. Mehdipour et al., in preparation).
An estimate of the AMD can be obtained assuming that each ion contributes its entire column at the where the ion’s relative ionic abundance peaks. The total equivalent for each is then estimated by each ion:
(7) 
This is a lower limit on column densities since in general . The estimate is plotted in Fig. 5, and shows a slight increase in column with consistent with Behar (2009). Different ions from different elements in the same bin should agree, and discrepancies reflect deviations from solar abundances.
In order to compute the AMD we want to solve the discretized set of equations (6)
(8) 
where is the matrix of ionic fractions given by XSTAR multiplied by , N is the vector of measured ionic column densities, and is the vector of H column densities we want to find multiplied by the vector of AMD bins. Note the AMD vector is rebinned manually and may be uneven, enlarging the size of the bin until significant constraints are obtained for each bin. The predicted columns are . We use Cstatistics (Cash, 1979) to fit the AMD as we expect zerovalue bins and there are less than 30 d.o.f. We minimize the in order to find a best fit for the AMD:
(9) 
The uncertainties of the measured ionic column densities are propagated stochastically. We use 1000 MonteCarlo runs on the vector N, where each column density is rolled from a triangular probability distribution ranging through the 90% confidence interval peaking at the best fit.
The resulting AMD is plotted in Fig. 6, and resembles the AMD of Blustin et al. (2007). This is also well in agreement with the usual bimodal shape commonly observed in AGNs (Behar, 2009; Laha et al., 2014). The consistency of the AMD structure along with the individual ionic column measurements increases our confidence that the absorber is unchanged between the 2004 and 2015 observations.
4 Variability and electron density
Following the previous work of Krolik & Kriss (1995); Nicastro et al. (1999), and Arav et al. (2012) we constrain a lowerlimit on distance of the source to the outflow using the fact that no variability is measured in ionic column densities. From this we can estimate upper limits on . In Appendix A a rigorous derivation of the equations used in this section is provided for reference.
4.1 Days time scale variability
The NGC 7469 lightcurve, created using the high statistics of the EPICpn, shows NGC 7469 has a variable continuum. In Fig. 2 the 9 EPICpn lightcurves are presented, two from 2004 and the rest from 2015, with the count rate varying by up to a factor of 2 within a day. This rapid variability (compare with the year time scales, Sec. 4.3) suggests the possibility of constraining the minimum response time to a change in ionizing flux of NGC 7469, and giving a lower limit on and thus an upper limit on the distance of the outflow from the AGN. This would only be possible if ionic column densities would be observed to change within the timescales of the continuum variability. In our case no variability can be detected on scales of days and longward, and thus only lower limits on distance and upper limits on may be obtained.
Since we can constrain the column densities at best to 50%, evaluated by comparing the uncertainties to the best fit values, weaker variability is not ruled out. Conversely, the lack of detected variability in individual ions, as well as a lack of a systematic trend in the discrepancies between bestfit values, implies that if any change exists, it is small and may not be attributed to a change of the ionizing flux. UV observations are more sensitive to variation in absorption troughs, and a detailed UV analysis of the epochs of NGC 7469 will be presented in a separate paper (Arav et al. in preparation).
In order to check the stability of the absorption due to the ionized outflow we apply the bestfit model on the combined spectra as a starting point for the fit of each individual spectrum. Though the lower S/N of a single observation hampers tight constraints, the results are consistent within the 90% uncertainty intervals across observations (Fig. 7), even better. The only exceptions are Ne and Fe deviating for one observation, but not the same one. Beyond constancy among observations, when considering the bestfit values it is evident that there is no clear trend  the ordering of column densities of different ions of similar ionization parameter between observations is not uniform. This indicates there is no observable change, in fact, of the ionic column densities during the last half year of 2015.
4.2 Intraday time scale variability: comparing high and low states
The stochastic nature of the ionizing flux may lead to an hypothesis that any outflow which is not dense and close to the source would not respond quickly enough to the changes, at least not measurably. By summing spectra of predominantly high and low states of the AGN separately, more subtle changes can be measured by improving the S/N of small absorption troughs which change in a consistent manner, on a daily basis.
We divide the states according to the EPICpn lightcurve, around the mean count rate for the 2015 observations (which is nearly identical to the median one) of 23.2 counts s. Retaining all photons in favor of statistics and in order to secure similar RGS S/N in the high and low states, we cut the events at above the mean EPICpn count rate ( counts s is the standard deviation of the light curve). These spectra are presented in Fig. 8, showing very similar troughs. As was the case for the individual epoch analysis, we begin a fit from the bestfit model of the combined spectra. Results are presented in Fig. 9. Here 3 ions only deviate, which is expected within the 90% statistics.
Once again the NGC 7469 outflow proves to be remarkably stable such that when observing only times of high flux and comparing to times of low flux, no change is observed in column densities and thus the outflow ionization distribution. Here, variability is constrained at best to 25% (by comparing the uncertainties to the best fit values), and again, weaker variations may be present.
4.3 Year time scale variability
While on timescales of days and less we see that the continuum of NGC 7469 is variable in EPICpn lightcurves (Fig. 2), ionic column densities remain unchanged over timescales of days and months, observed during 2015. In addition, the column densities are comparable to those of 2004, despite the 25% difference in flux (See Table 2). Thus, we make the assumption that column densities remain unchanged for the entire years. This assumption allows us to constrain the distance of the outflow from the ionizing source assuming , where is the ionization equilibrium time (Krolik & Kriss, 1995; Nicastro et al., 1999; Arav et al., 2012).
A full derivation of the dependence of on is detailed in Appendix A. While the power of this derivation cannot be fully utilized for NGC 7469 as we detect no variability, a useful result for this case is:
(10) 
Given an ion the recombination rate coefficient is . The photoionization crosssection and rate are, respectively, and , where
(11) 
is the luminosity density in erg s keV and is the ionizing AGN luminosity:
(12) 
and are estimated from the SED, which yields erg s. Recombination and photoionization coefficients are taken from the CHIANTI software package^{4}^{4}4http://www.chiantidatabase.org/ (Landi et al., 2013). Once we obtain a lower limit on the distance, we may use the definition of to extract an upper limit on :
(13) 
where is the minimal value obtained from eq. 10, and the same and values are used.
Distances and electron densities measured from several ions are given in Table 4. The outflow is constrained to be at least 12 pc away from the source for Fe and 31 pc for N. This constraint is not strong enough to disassociate it from the AGN completely, or associate it with the starburst region seen in NGC 7469 (David et al., 1992), which is approximately 1 kpc from the source.
ion  

erg cm s  cm s  cm s  cm s  pc  cm  
N  
O  
Fe 
5 Energy deposit
Using mass conservation in a continuous conical outflow with opening angle , , we define the kinetic power of the outflow as
(14) 
Where is the mean molecular weight and is the proton mass. We assuming here a biconical flow of . We use the maximal velocity component of km s and the lowest ionization observed at that velocity, (excluding ions with column density consistent with 0). These values are seen for example in C and O. This yields a maximal possible value of
(15) 
where erg s. Using a high but leaving the velocity of km s (observed for example in Fe) will reduce this value by two orders of magnitude:
(16) 
Substituting in the lowest velocity of km s will reduce by another 1.5 orders of magnitude, and an opening angle less than would reduce it even further.
The fact that a range of values is ubiquitously observed in AGN outflows indicates the wind cannot have a conical density profile. Multiple ionization winds have been discussed in the models of Fukumura et al. (2010); Stern et al. (2014). Eq. 14 results in an increase of power with decreasing ionization.
Other definitions of kinetic luminosity, such as that of Borguet et al. (2012), assume a thin shell of thickness rather than a continuous outflow, dividing the mass by the traversal timescale, . In that case the kinetic luminosity would be lower by .
One may also assume , such that . In this case we can use the measured lower limits on distance (Table 4) and the measured equivalent H column densities (Eq. 7). Lower bounds for from N, O, Fe respectively are , and erg s. Note for each ion we use the fastest velocity where measured column density is inconsistent with 0, namely –600 km s for N, and –2000 km s for O and Fe. The lowest estimate is even lower than that of Equation 16.
Assuming the highest estimate of the kinetic power (Eq. 15) is the true energy carried by the outflow would imply significant feedback. However, the fact is that a starburst region is observed at 1 kpc (David et al., 1992) and does not seem to be affected by the outflow. This would lead to the conclusion that the outflow is spatially decoupled from the starburst region. If the outflow power is much lower as in Eq. 16, this would naturally explain why the starburst region is unaffected.
6 Conclusions
The Xrays absorption spectra of NGC 7469 is remarkably stable on all of the measured time scales. In observations spread over years, months and days column densities associated with the ionized absorber are not observed to change. On the other hand, the intrinsic variability of the source is large, changing by up to a factor of two in the course of a single day. In addition, the average soft Xray powerlaw slope changes between 2004 and 2015 from 2.1 to 2.3, again, with no observed absorption variability.
The kinematic components of the outflow are also constant between the 2004 and 2015 observations, and between the Xray and the UV bands. Constancy of the outflow can also be observed in the reconstructed AMD, featuring one high ionization component and one low ionization component with the same column densities in both 2004 and 2015. Admittedly, the broad and relatively flat AMD makes ionization changes much harder to detect than in a single component. To that end, we would expect to notice changes only in the highest and lowest ionization states. Nonetheless, the UV spectra of this campaign (Arav et al. in preparation) confirms for the most part the lack of absorption variability, except for minor changes that are detected in a few velocity bins in the UV, but are much below the current Xray sensitivity.
The flux variations on different timescales with no effect whatsoever on the outflow imply a distant outflow, several pc away from the AGN at least. Beyond the large distance, the velocities, luminosity, and observed ionization parameters suggest the outflow may carry as much as 2/3 of the Eddington AGN power, which is significant in terms of feedback. However, this is dependent on (Eq. 14) as expected for nonconical outflows, and is 2 orders of magnitude lower for high values, making these estimates ambiguous and inconclusive as estimators of feedback without a physical model associated with .
We found no evidence the AGN is responsible for driving the outflow, since the distance scales are beyond the torus (Suganuma et al., 2006) and comparable to the region of narrow ( km s) line emission. The obtained constraints on distance and power of the outflow need to be examined in other AGNs in order to understand if these outflows are unimportant to the galactic scale, and what is their connection to the AGN itself.
Acknowledgements.
This work was supported by NASA grant NNX16AC07G through the XMMNewton Guest Observing Program, and through grants for HST program number 14054 from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS526555. The research at the Technion is supported by the ICORE program of the Planning and Budgeting Committee (grant number 1937/12). EB received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie SklodowskaCurie grant agreement no. 655324. SRON is supported financially by NWO, the Netherlands Organization for Scientific Research. NA is grateful for a visitingprofessor fellowship at the Technion, granted by the Lady Davis Trust. SB and MC acknowledge financial support from the Italian Space Agency under grant ASIINAF I/037/12/0. BDM acknowledges support from the European Union’s Horizon 2020 research and innovation programme under the Marie SkłodowskaCurie grant agreement No. 665778 via the Polish National Science Center grant Polonez UMO2016/21/P/ST9/04025. LDG acknoweledges support from the Swiss National Science Foundation. GP acknowledges support by the Bundesministerium für Wirtschaft und Technologie/Deutsches Zentrum für Luft und Raumfahrt (BMWI/DLR, FKZ 50 OR 1408 and FKZ 50 OR 1604) and the Max Planck Society. POP acknowledges support from CNES and from PNHE of CNRS/INSU.References
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Appendix A Equilibrium time
Following the works of Krolik & Kriss (1995); Nicastro et al. (1999); Arav et al. (2012) we define the two inverse timescales for ionization and recombination respectively:
(17)  
(18) 
The ionization/recombination/equilibrium time used in this paper is the decay time of the exponential solution of the system of equations for the ionic populations :
(19) 
for charge states and the boundary defined by or:
(20)  
(21) 
These equations assume all charge states are exposed to the same radiation field . In the general case where the radiation field is nonuniform this approximation breaks down.
a.1 Assumptions and caveats
In general, 19 must be solved for a time varying set of , making the full solution much more difficult, and is formally given in Krolik & Kriss (1995). This is less practical when we want to use our measurements to constrain unobserved quantities, such as . In this case, we often want to consider a system in equilibrium, with a given inital set , where we abruptly change the external conditions using a new set of  making the assumption that the continuum changed as a step function, and we are observing much after the step (See Sec. A.3), or conversely that the system is in equilibrium and this abrupt change has yet to be observed. Though often not the case, this is a good assumption when observing the outflow much before and much after such a change in seed flux, such that the continuum observed is steady for times greater than the . Consider now a short scale oscillating variation in seed flux,
(22) 
AGNs in general (indeed, NGC 7469 is a good example) may change drastically on timescales of days, with no observable change in column densities. In this case we may assume that the effective continuum on the plasma is in fact a steady one, given by the time averaged flux.
(23) 
So we make 3 assumptions when analyzing this photoionized plasma:

If no column densities are observed to change while flux varies on short (hours,days) time scales, a steady time averaged continuum may be assumed.

If column densities are changed between two observations, and the flux is shown to be steady, we will assume where is the final observation and is the time where the continuum started to change, after the first observation. In this case we assume a step function change for the continuum.

Finally, if column densities remain unchanged between two observations but flux is shown to have changed and remain steady, we will assume , the time between observations.
a.2 Solution
From the form of the equations, or from solving the simple 2level system one may quickly come to the conclusion a general solution should be of the form (assuming constant , as per Sec. A.1):
(24) 
Firstorder differential equations have only one free coefficient depending on the initial conditions. must be independent of charge, and this can easily be shown by substituting different for consecutive charge states into the equation for , 19, assuming and are constants. Some properties of this solution are evident immediately. Assuming steady state before and at leads to the conclusion:
(25)  
(26) 
where are the equilibrium densities at and are the initial equilibrium densities. An important consequence is that are not integration coefficients. These are the final equilibrium solutions, explicitly given by , as seen in the Section A.5. Substitute in our form 24 to 19:
Grouping the coefficient for the exponent and constant results in the formulas for the coefficients:
(27)  
(28) 
It is easy to prove that 28 results in which we know must be true, as are an equilibrium solution (see Sec. A.5). What will be interesting to us is the relation of to .
a.3 Equilibrium time
A closed form solution for is more difficult, but we are only interested in , which may be obtained from 27 using any observed ionization triad:
(29) 
Measuring 3 ions of an element and seed flux of 2 different observation epochs will allow us to constrain . In terms of what we measure:
(30) 
where are the column densities of the specific ions and is the ratio of widths over which the two ions extend. We will assume as is inversely proportional to and Kallman et al. (1996) shows most adjacent ion stages tend to extend over similar ranges, and indeed may exist in the same part of the plasma, though this does not have to be the case.
An interesting thing to note is that the equilibrium constants are also dependent on the and , and obviously each is a different set of constants as both and have changed, but only those of are the same as the explicit and appearing in 29. Finally, substituting the expressions for obtain the relationship we need:
(31) 
We note this equation is the same as eq. 10 in Arav et al. (2012) when
(32) 
and , tying a step change in ionization flux to recombination.
Note that the ionization parameter is an observable that is found independently:
(33) 
While at first glance this may seem like it would be embedded somehow in 31, note that is a purely equilibrium characteristic of the plasma, while is of course the time scale characterizing the system out of equilibrium. This gives us physical justification to say 31 and 33 are independent equations, and may be solved simultaneously for and :
(34) 
(35) 
An interesting consequence is that the coefficients of must be positive. If this is not the case, then these solutions are wrong and our assumptions need to be put to test. Note that for a two level system this must be true as the ratio of column change is always negative.
a.4 Applications
While most parameters insofar are either measurable independently () or known () we in general only have a limit on as we do not observe the plasma continuously. To practically apply this result to observational data we need inequalities, not equalities. Assume we know is lower than some constant , a time between two observations. This happens often when we see an AGN in a steady low/high state at one time, and a high/low in another, with different columns. We can then use:
(36) 
(37) 
to constrain a maximal , and minimal electron density. If on the other hand no variability is measured we are struck with a problem. While we would know , so constraints would be reversed, we do not know the final column densities. One way to handle this is to make the assumption is, as a two level system, always negative, allowing an estimate:
(38) 
and consequently following from eq. 33 we have:
(39) 
where is obtained from the lower limit given by eq. A.4. This is the approximation used in Sec. 4.3.
a.5 Equilibrium
We add this section for completeness’ sake only. This problem can trivially be solved for the case , where by induction if and
(40) 
Substituting in the induction assumption we have the well known result:
(41)  
(42) 
This is easy to show for the first pair using 20=0. This recursive solution is quickly generalized for the relationship between and , where and respectively:
(43)  
(44) 
Finally we note that our system when summed is telescopic, that is:
(45)  
(46) 
where we have defined as the constant number of particles. Thus we obtain a complete closed form solution, starting from equation 46 and solving for :
(47) 