# Presentation: 801753

## Emission II:Collisional Plasmas

- Randall K. Smith
- Chandra X-ray Center

## Introduction

- We have covered the basic atomic processes that are important in X-ray emitting plasmas: collisional excitation/ionization, photoexcitation/ionization, radiative decay and so on.

- X-ray emitting plasmas are separated into two types:
-  Collisional:     kBTe ~ Ionization energy of plasma ions
-  Photoionized: kBTe << Ionization energy of plasma ions

- What about plasmas in local thermodynamic equilibrium (LTE)?

- This occurs if Ne > 1.8 x 1014 Te1/2 Eij3 cm-3.

- For Te=107K for H-like Iron, Ne > 2x1027 cm-3.

- For Te=105K for H-like Oxygen, Ne > 1024 cm-3.

## Introduction

- Astrophysical **collisional** plasmas come in two types:

-  Coronal/Nebular:          Ne < 1014 - 1016 cm-3
-  Collisional-Radiative:  1014 cm-3 <  Ne < 1027 cm-3

- In a CR plasma, collisions compete with photons in de-exciting levels; a level with a small oscillator strength (transition rate)  value may be collisionally de-excited before it can radiate.

- In the more common Coronal (or Nebular) plasma, collisions excite ions but rarely de-excite them; the decay is radiative.  This is also called the “ground-state” approximation, as all ions are assumed to be in the ground-state when collisions occur.

## Introduction

- We will make some initial assumptions about our “astrophysical plasmas”:
-  They are dominated by H and He, with trace metals.
-  Any magnetic and electric fields do not significantly affect the ion level structure.
-  No nuclear transitions are significant.
-  The electrons have a thermal (Maxwellian) velocity distribution.

## Optical Depth

- But what about radiative excitation?  Can’t photons still interact with ions, even in a collisionally ionized plasma?

## Optical Depth

- So, is photon scattering an important process?

- Yes, but only for allowed transitions; in a collisional plasma, many transitions are forbidden or semi-forbidden.

- So couldn’t this show up as optical depth in allowed lines, weakening them relative to forbidden lines?

- Yes, and this can be calculated after modeling a plasma.  Using the ionization balance and the coronal approximation, along with the A value for the transition and the emitting volume, it is easy to calculate the optical depth for a line:

-  = nI  l

- This effect is often not important, and even less often checked!

## Equilibrium

-  A collisional plasma in ionization equilibrium (usually called a CIE plasma) has the property that
- Irate(Ion) + Rrate(Ion) = Irate(Ion-) + Rrate(Ion+)
-  A non-equilibrium ionization (NEI) plasma may be:
-  Ionizing  [Irate(I) >  Rrate(I)]
-  Recombining  [Irate(I) <  Rrate(I)]
-  Other

- Both CR and Coronal plasmas may be in equilibrium or out of it.

## Equilibrium

- The *best* term to describe the topic of this talk is:
- optically-thin collisional (or thermal) plasmas

- Frequently, the “optically-thin” portion is forgotten (bad!)

- If the plasma is assumed to be in equilibrium, then CIE is often used, as are phrases like:
-  Raymond-Smith
-  Mekal
-  Coronal plasma (even for non-coronal sources...)

- Out of equilibrium, either NIE or NEI are used frequently, as are:

-  Ionizing
-  Recombining
-  Thermal + power-law tail

## Spectral Emission

- So what do these plasmas actually look like?

- At 1 keV, without absorption:

## Bremsstrahlung

- Was ist der Bremsstrahlung?
- First seen when studying electron/ion interactions. Radiation is emitted because of the acceleration of the electron in the EM field of the nucleus.

- Importance to X-ray astronomy:
-  For relativistic particles, may be the dominant coolant.
-  Continuum emission shape dependent on the e- temperature.
-  Ubiquitous: hot ionized gas Bremsstrahlung radiation.
- The complete treatment should be based on QED, but in every reference book, the computations are made “classically” and  modified (“Gaunt” factors) to take into account quantum effects.

## Bremsstrahlung

- Non-relativistic: Uses the dipole approximation (fine for electron/nucleus bremsstrahlung)

- -e

- Ze

- v

- b

- The electron moves mainly in straight line--

- R

- And the electric field is:

## Bremsstrahlung

- Now use a Fourier transform to get:

- And the emitted energy per unit area and frequency is:

- Integrating over all solid angles, we get:

## Bremsstrahlung

- Consider a distribution of electrons in a medium with ion density ni, electron density ne and constant velocity v.  Then the emission per unit time, volume, frequency:

- Approximate this by considering only contributions up to bmax and integrating:

- where

- and

## Bremsstrahlung

- Karzas & Latter, 1961, ApJS, 6, 167

- The full QED solution is:

- gff

- log E/Z2 (Rydbergs)

- gff = 1 line

## Bremsstrahlung

-  where <gff> is the velocity average Gaunt factor

- Now integrate over electrons with a Maxwell-Boltzmann velocity distribution:

- To get:

## Bremsstrahlung

- From Rybicki & Lightman Fig 5.3 -- originally from Karzas & Latter (1961)

-  u =h/kT ;
- 2 = Ry Z2/kT
-     =1.58x105 Z2/T

- Numerical values of <gff>.

- When integrated over frequency (energy):

## Global Fitting

- CCD (or proportional counter) data are regularly fit in a global fashion, using a response matrix.  If you believe that the underlying spectrum is from an optically-thin collisional equilibrium plasma, then you can “fit” your choice of collisional plasma model (apec, mekal, raymond, equil are available in XSPEC or sherpa).

- By default, the only parameters are temperature and emission measure.  If the fit is poor (2/N > 1) you can add more parameters: such as the overall abundance relative to solar, or the redshift.

- If the models are still a poor fit, the abundances can be varied independently, or the equilibrium assumption can be relaxed in a few ways.

## Global Fitting

- Are there problems with this method?

- Of course there are.  However, when your data has spectral data has resolution less than 100,  you cannot easily identify and isolate X-ray spectral lines -- but low resolution data is better than no data: the goal is understanding, not perfection.

-  If the underlying model is inadequate, your results may be as well.  Beware especially abundances when only one ionization state can be clearly seen.
-  Cross-check your results any way you can.  For example, the EM is related to the density and the emitting volume.  Are they reasonable?
-  If you can’t get a good fit in a particular region, your problem may be the model, not the data.

- It is vital to keep in mind:

## Global Fitting

- Consider this ASCA CCD spectrum of Capella, with a collisional plasma model fit:

## Global Fitting

- In this case, the poor fit between 9-12 Å is likely due to missing lines, not bad modeling.

## Global Fitting

- Here is a parallel shock (pshock, kT=0.7 keV), observed with the ACIS BI:

- An NEI collisional model fits the data quite well.

- But with higher resolution...

- the NEI model fails, pshock is needed.

- O VII

## NEI vs CIE Emission

- We can compare a CIE plasma against an NEI plasma, in this case an ionizing plasma, also at 1 keV.

## Ionization Balance

- In order to calculate an emission spectrum the abundance of each ionization state must be known.  Shown here are four equilibrium ionization balance calculations for 4 iron ions:

## Ionization Balance

- In some cases, the differences are small.  Here is a comparison of O VI, VII, VIII, and fully-stripped Oxygen, for three different models:

## Ions of Importance

*All ions are equal...*

*...but some are more equal than others.*

- In collisional plasmas, three ions are of particular note:

**H-like** : All transitions of astrophysically abundant metals (CNi) are in the X-ray band.  Ly/Ly is a useful temperature diagnostic; Ly is quite bright.

**He-like**: n≥1 transitions are all bright and in X-ray.  The n=21 transitions have 4 transitions which are useful diagnostics, although R=300 required to separate them.

**Ne-like**: Primarily Fe XVII; two groups of bright emission lines at 15Å and 17Å; ionization state and density diagnostics, although there are atomic physics problems.

## Ions of Importance

- Fe XVII

- Ne IX

- O VIII

- O VII

- Ne X

- Capella observed with the Chandra HETG

## Hydrogenic Lines

- Three calculations of the O VIII Ly line as a function of temperature.

## Hydrogenic Lines

- Three calculations of the O VIII Ly/Ly line as a function of temperature (APEC agrees with measurements).

## Helium-like Lines

- One useful He-like diagnostic is the G ratio, defined as (F+I)/R  [or, alternatively, (x+y+z)/w].  It is a temperature diagnostic, at least for low temperatures, and it is also measures ionization state.

## Helium-like Lines

- Why does the G ratio measure temperature and ionization state?

- Because the resonance line R is excited by collisions, which are temperature dependent, while the F and I lines are excited by recombination and other processes.

**G = (F+I)/R**

## Helium-like Lines

- The ratio F/I is normally called the R ratio, and it is a density diagnostic.  If ne is large enough, collisions move electrons from the forbidden to the intercombination and resonance levels.

## Helium-like Lines

- How well are these He-like lines known?  Here are three calculations for each of the three lines:

## Neon-Like Lines

- Fe XVII is the most prominent neon-like ion; Ni XIX is 10x weaker simply due to relative abundances.  There are a number of diagnostic features, as can be seen in this grating spectrum of the WD EX Hya (Mauche *et al*. 2001):

## Neon-Like Lines

- Here they have extracted the ratio of two very closely spaced Fe XVII lines, which are a density or a UV flux diagnostic

## Neon-Like Lines

- What about the strong 15.02Å and 15.26Å lines?

- They should be useful diagnostics, but right now we’re still debating their proper ratio...stay tuned

- Bhatia & Saba 2001

## Plasma Codes

- Understanding a collisional plasma requires a collisional plasma model.  Since even a simple model requires considering hundreds of lines, and modern codes track millions, most people select one of the precalculated codes:

| Code
Raymond-Smith
SPEXChianti
ATOMDB | Source
ftp://legacy.gsfc.nasa.gov/software/plasma_codes/raymond
http://saturn.sron.nl/general/projects/spex
http://wwwsolar.nrl.navy.mil/chianti.html
http://cxc.harvard.edu/ATOMDB |
| --- | --- |

## Plasma Codes

- Individual line intensities as functions of T, n, etc. are not easily available (yet) in either XSPEC or Sherpa.

- The collisional plasma models available in XSPEC or Sherpa are:

| apec
raymond
meka
mekal
c6mekal
equilnei
sedov
pshock | ATOMDB code; good for high-resolution data
Updated (1993) Raymond-Smith (1977) code
Original Mewe-Kaastra (Mewe et al. 1985) code; outdated
Mewe-Kaastra-Liedahl code (Kaastra 1992); new Fe L lines
mekal with an polynomial EM distribution
Borkowski update of Hamilton, Sarazin & Chevalier (1983)
Ionizing plasma version of equil
Sedov (SNR) version of equil
Plane parallel shock version of equil |
| --- | --- |

- Variable abundance versions of all these are available.

## Atomic Codes

- HULLAC (Hebrew University / Lawrence Livermore Atomic Code) : Fast, used for many APED calculations, not generally available.

- R-Matrix : Slow, used for detailed calculations of smaller systems of lines, available on request but requires months to learn.

- FAC (Flexible Atomic Code) : Fast, based on HULLAC and written by Ming Feng Gu.  Available at

- ftp://space.mit.edu/pub/mgfu/fac

## Conclusions

- So you think you’ve got a collisional plasma: what do you do?

-  If high resolution data are available, line-based analysis allows the best control of errors, both atomic and data/calibration.
-  If CCD (or worse) is all that you have, remember Clint Eastwood’s admonition:

- A spectroscopist’s gotta know their limitations.

- Keep in mind that :
- only the strongest lines will be visible,
- they could be blended with weaker lines,
- plasma codes have at least 10% errors on line strengths,
- the data have systematic calibration errors, and finally:
- the goal is understanding, not 2n ~ 1 fits.

## Bremsstrahlung

- From Rybicki & Lightman Fig 5.2 (corrected) -- originally from Novikov and Thorne (1973)

- Approximate analytic formulae for <gff>

**Notes:**
Note that at hnu/kt <<1 G only depends on the log of frequency-- (small variation). 
X-ray == g(nu, T) = sqrt(3)/pi ln(kt/hnu)