4.1.28. Pion: SPEX photoionised plasma model

The pion model calculates the transmission and emission of a slab of photo-ionized plasma, where all ionic column densities are linked through a photoionisation model. The relevant parameter is the ionization parameter \xi = L/nr^2, with L the source luminosity, n the hydrogen density and r the distance from the ionizing source. The major difference is that while for the xabs model (Xabs: photoionised absorption model) the photoionisation equilibrium is pre-calculated for a grid of \xi-values by an external code, for instance Cloudy or XSTAR (or even with the present pion model \ldots), in the present pion model the photoionisation equilibrium is calculated self-consistently using the available plasma routines of SPEX.

Warning

The default energy grid in SPEX has 8192 bins between 0.001 and 100 keV. This may not be sufficient for the pion model when you use it without data. Recommended is a logarithmic grid between 10^{-6}-10^{6} keV with a step size of 0.005. You can get this by issuing the following command: ”Egrid log 1E-6 : 1E6 step 0.005 keV”. Note that if you have read in data, SPEX automatically uses the resolution of the response matrix within its energy range and expands the total energy grid to 10^{-6}-10^{6} keV.

Warning

This model is still under development and not all atomic data is fully updated. For instance, no cooling by collisional excitation for ions of the K- to Zn-isoelectronic sequences is taken into account yet. So use with care!

Warning

When setting up the model, be aware that the pion model is both additive and multiplicative (even if you put the emission to zero). Therefore, the pion model needs the same com rel sequence as you use for your absorption component. Example: com powcom redscom pioncom rel 1 3,2 (a powerlaw that powers a pion model and is then redshifted) needs as next command com rel 3 2, telling SPEX that the pion emission model is also redshifted.

Warning

Please note that all PION components must be multiplied by at least one continuum component (using the usual comp rel command). Otherwise, photoionisation cannot be calculated and SPEX may crash. A typical AGN SED spanning optical to X-ray energies can be set up with three continuum components in SPEX: pow (primary hard X-ray power-law emission), refl (reflected power-law emission), comt (thermal optical/UV disk continuum + the soft X-ray excess). See the SED of NGC 5548 derived in Mehdipour et al. (2015).

Warning

Starting from SPEX Version 3.06, we have updated the cooling due to collisional excitation (work performed by Stofanova et al., 2021, submitted). In addition, starting from SPEX Version 3.06.01, we have improved the collisional excitation rates of neutral hydrogen; the latter affects both emitted H I spectra, and the total cooling rate at the lower temperatures (few eV). That cooling rate may differ at some temperatures by a factor of 3-4. Therefore results obtained with the most recent version will differ from those of version 3.05 and earlier.

Warning

When using the pion model with the ion mute/unmute option in order to hide or show the contributions from specific ions to the total spectrum, you will get a different solution, because it affects the heating and cooling rates (muted ions will not contribute to the heating and cooling in the calculations), and thus the ionisation balance (equilibrium temperature) will change. Exception is when you use the tmod=1 option for pion, which forces the temperature to be equal to what you prescribe through parameter tinp. For diagnosing the heating/cooling contributions of ions or elements, it is therefore recommended to run first the model with all ions, make an ascii-output of the plasma parameters, take the temperature from there as the “tinp” parameter for the pion model, and set tmod=1. You can play then with the mute/unmute command.

The main advantage, however, of the pion model is that the user can define his own ionising continuum by combining any additive components of SPEX, and that ionising spectrum is fed through the pion component to get the transmitted spectrum. In addition, multiple pion components can be used, with the light leaking from one layer to the next. And it allows for spectral fitting, where the parameters of the ionising continuum are determined simultaneously with the parameters of the absorbing layer. The number of parameters is therefore unlimited (well, the memory and cpu speed of the computer will be the true limiting factors).

Prior to fitting a PION component, it is best to first calculate the model spectrum with your initial parameter values (see Calculate: evaluate the spectrum). This helps to see if the initial values are reasonable numbers and the model is not too far off the data.

4.1.28.1. Emission from the pion model

We have now incorporated a first version of emission from the same layer. We cannot give you any guarantee at the moment that it is bug-free. We know at the moment that the source has problems when the density gets too high; this is different for each ion; so unless you limit the density in fitting, SPEX may encounter a situation where you surpass the critical density and you may get a warning message. Here is an example. For a photoionised case, with \log \xi = 3 (resulting kT=0.64 keV) the nominal occupation of the ground state of H becomes negative for a density >0.15\times 10^{20} \mathrm{m}^{-3}. This can be traced down to incomplete atomic data. For H, we include collisional excitation and de-excitation up to principal quantum number n=5 but not above. As a result, in this example the 1s–5s levels are mainly populated/depopulated by collisions, while 6s–16s are mainly populated by radiative recombination and depopulated by radiative cascades downward or photon absorption. The nominal occupation of 1s–5s then decreases from 0.035 (1s) to 0.0024 (5s), while for 6s–16s they increase slowly from 0.020 to 0.027. This is of course physically unacceptable. It causes the lower levels to leak to the higher levels, with eventually the catastrophe of negative occupation for the ground state. We mitigate this by replacing the level populations by the LTE populations and issueing a warning. Without this mitigation, SPEX would crash.

Warning

You can get the emission by putting the covering factor (omeg) to a non-zero value; it will slow down the calculations compared to absorption-only calculations, so be aware.

Normally, to calculate the emission from a full thin shell surrounding an ionising source, you should set the parameter omeg to unity (a full shell of 4\pi steradians). Smaller factors could be associated e.g. to ionisation cones; values larger than unity make physically no sense but you can formally play around with it (for very large values, the emitted spectrum would start dominating the absorbed primary continuum, but if you want to suppress the primary continuum in the observed spectrum, it is better to define your model like the example below as:

SPEX> com pow
SPEX> com pion
SPEX> com etau
SPEX> com rel 1 2,3
SPEX> com rel 2 0
SPEX> par 3 tau v 1e10
SPEX> par 3 a v 0

In this example, the powerlaw goes through the pion component and is killed afterwards by the etau component (Etau: simple transmission model), while the emission from the pion component is not attenuated by etau.

You can vary the new parameter mix to get a different ratio of forwards to backwards emission. Putting it to 1 (default) means you get the forward emission, putting it to 0 the backwards emission, and intermediate values give you a mix.

Warning

The emission model uses currently only one layer. When the continuum optical depth of the absorbed continuum, weighted with the incoming flux, becomes of order unity, the layer will become inhomogeneous in terms of temperature structure, and our single-layer approximation will break down.

In order to make a pion component produce emission only, fix the covering fraction (cf) parameter to zero so that no absorption is produced. Then fit the omega parameter. Note that any pion component with a non-zero omega acts as an additive component in SPEX. Therefore, multiply these components with your multiplicative components (like the Galactic absorption) using the comp rel command.

For more information on this model, the atomic data and parameters we refer to Different types of absorption models.

4.1.28.2. More options

4.1.28.2.1. No energy balance solution needed

The default option (tmod=0) for the pion model is to solve simultaneously for the ionisation balance and the energy balance equations. This option is useful for e.g. photoionised winds of AGN or X-ray binaries.

However, there are situations where photo-ionisation or photo-excitation play a role but do not determine the thermal structure. Examples are winds of hot stars, where shocks heat the wind but UV radiation from the star can affect He-like triplet line emission ratios. Another example are the most teneous parts of the WHIM, where photoionisation by the cosmic background can be important compared to collisional ionisations.

For such cases, the user can set the parameter tmod=1; in that case, the user should also provide the temperature tinp of the plasma. In this case, only the ionisation balance equation is solved, and there is in general no energy balance (this can be checked by using the ascii-output option heat, Ascdump: ascii output of plasma and spectral properties). Do not forget to set the parameter omeg to a finite value (the default is zero), otherwise the emitted spectrum is zero.

4.1.28.2.2. External heat sources

In some cases there may be an other external heat or cooling source, like shock heating, magnetic reconnection, adiabatic expansion etc. If one wishes to solve for the photoionisation equilibrium, then this additional heat source can be used by putting the parameter exth to the proper value (units: W \mathrm{m}^{-3}. A negative value would mean a cooling contribution.

4.1.28.2.3. Multiple solutions

There are situations where there is not a unique solution to the energy balance equations. A simple example can be obtained as follows: take a logarithmic energy grid between 10^{-6}-10^6 keV, use a powerlaw with photon index 1.5, apply the pion model to it and put exth to 5\times 10^{-25} W \mathrm{m}^{-3}. In this case there are 3 solutions. SPEX chooses by default the hottest solution. You can see all solutions by putting the parameter fmod=1 and using the heat ascii output option. Or check the behaviour of the heating balance by issuing the ebal ascii output option (Ascdump: ascii output of plasma and spectral properties). You can select which solution you want to use in SPEX by setting the soln parameter. Default is 0 (hottest solution), and for the above case of 3 solutions values of 1, 2 and 3 renders you the coldest, second ant hottest solution. Test this with the heat or plas output options.

Warning

When you set soln to a non-zero value, use fmod=1, otherwise SPEX may crash.

4.1.28.2.4. No equilibrium solution

There are also situations where there is no equilibrium solution to the energy balance equations. This may happen for instance if you put so much heat in the plasma that it cannot be balanced anymore by cooling. Another example is a too hard powerlaw without high energy cut-off, where Compton-heating might be very strong. In this case SPEX renders an error message, and you cannot trust the result of the calculation anymore. The only remedy is to adjust your model parameters or the allowed range for them in case of spectral fitting or error searches.

4.1.28.2.5. Adiabatic cooling

The effects of adiabatic cooling can be taken into account by setting the parameter tadi. This represents the adiabatic cooling time t_{\mathrm adi}. The associated cooling rate is calculated as R_{\mathrm adi}= \frac{3}{2} nkT / t_{\mathrm adi}, where n is the total particle density (electrons and ions). The default setting is such that this process can be neglected. If the user takes this process into account, it should be verified afterwards that the physical conditions for adiabatic cooling are met, i.e. energy losses by radiation or heat conduction must be small compared with those by the adiabatic expansion. Check this for example by running the asc ter ... heat output.

4.1.28.2.6. Radiative acceleration

The radiative acceleration caused by the absorption or scattering of the incoming radiation on the layer is calculated, and given as output parameter acc. Physically, it is given by the following equation, which can be easily derived:

a =F_{\mathrm abs}/c f m_{\mathrm p} N_{\mathrm H},

where F_{\mathrm abs}=\int F(E)(1-T(E)){\mathrm d}E is the absorbed flux (F(E) is the incoming flux in W \mathrm{m}^{-2} keV and T(E) the transmission of the layer), c the speed of light, m_{\mathrm p} the proton mass, N_{\mathrm H} the hydrogen column density and f is a dimension less quantity determined from \rho = f n_{\mathrm H} m_{\mathrm p} with n_{\mathrm H} the hydrogen density and \rho the mass density (kg \mathrm{m}^{-3}) of the plasma, for example f=1.4287 for the present default abundances of SPEX (you can check this number from the asc ter ... plas ascii output option).

Warning

It is important to note that the acceleration is proportional to the hydrogen density, so take care to choose the appropriate hydrogen density, even in the low density limit where the spectral shape does not depend on the density. This counter-intuitive behaviour is caused by the use of \xi as main parameter. Given the absolute ionising luminosity L of the ionising source, and the value of \xi and n_{\mathrm H} provided by the user, the pion model then calculates the distance r from the equation \xi = L/nr^2. Thus, higher density yields smaller distance, hence larger absorbed flux by the gas layer, hence stronger acceleration.

4.1.28.3. Model parameters

The parameters of the model are:

nh : Hydrogen column density in 10^{28} \mathrm{m}^{-2}. Default value: 10^{-4} (corresponding to 10^{24} \mathrm{m}^{-2}, a typical value at low Galactic latitudes).
xi : the ^{10}\log of the ionisation parameter \xi in units of 10^{-9} W m. Default value: 1.
u : the Davidson (Cloudy) ionisation parameter U (dimensionless). This is calculated from the SED and the value of \xi. Not fittable, just output.
hden : the hydrogen density in units of 10^{20} \mathrm{m}^{-3}. Default value: 10^{-14} (corresponding to 10^{6} \mathrm{m}^{-3}, to be consistent with the order of magnitude of the electron density (which is used in the cie and neij models; do NOT confuse both quantities!).

The following parameters are common to all our absorption models:

fcov : The covering factor of the absorber. Default value: 1 (full covering)
v : Root mean square velocity \sigma_{\mathrm v}
rms : Rms velocity \sigma_{\mathrm b} of line blend components
dv : Velocity distance \Delta v between different blend components
zv : Average systematic velocity v of the absorber
The following parameters are the same as for the cie-model (see there for a description):
ref : Reference element
01...28 : Abundances of H to Ni; only here we take H, He, C, N, O, Ne, Na, Mg, Al, Si, S, Ar, Ca, Fe, Ni.
file : File name for the electron distribution (in case of a sum of Maxwellians distribution)
The following parameters are unique for the pion model: type : If type equals 0 (default value), it uses \xi as its main parameter; if type equals 1, it uses lixi (see next parameter) as its main parameter
lixi : Optional alternative ionisation parameter, defines as L_{\mathrm
{ion}}/\xi in units of 10^{39} \mathrm{m}^{-1}. This is useful for time-variable spectra where \xi has been determined from one spectrum and where one wants to calculated the transmitted spectrum for fixed nr^2 for a different ionising spectrum; in that case lixi can be kept constant.
omeg : Covering factor \Omega/4\pi, needed for emission. At this stage, keep it to zero, please.
mix : Fraction of emitted spectrum to the forward direction relative to the total. default value: 1 (all emission forward). A value of 0 means SPEX gives all backwards emission.
exth : External heating in W \mathrm{m}^{-3}. default value: 0.
fmod : Show all solutions in ascii output of the heating (fmod=1). Default is fmod=0. Set fmod=1 also when you set soln>0.
soln : The temperature solution to be used, from low to high values. Default value is 0 (hottest solution). If this parameter is larger than the hootest solution, it adopts the hottest solution instead. Should be used with fmod=1 in case soln>0.
tmod : Temperature mode. Default value: 0 (solve for the temperature that provides energy balance). If tmod=1, use tinp instead as temperature and do not solve for energy balance.
tinp : Temperature of the plasma in keV. Default: 1 keV. Only relevant if tmod=1.
tadi : Adiabatic cooling time scale (s). See description above. Default value: 10^{30} s.
acc : Radiative acceleration. See description above. Note: only output.

Recommended citation: Mehdipour et al. (2016) and Miller et al. (2015)