Line absorption of He-like triplet lines by Li-like ions. Caveats of using line ratios of triplets for plasma diagnostics
aa r X i v : . [ a s t r o - ph . H E ] J un Astronomy&Astrophysicsmanuscript no. missagh˙mehdipour c (cid:13)
ESO 2018January 1, 2018
Line absorption of He-like triplet lines by Li-like ions
Caveats of using line ratios of triplets for plasma diagnostics
M. Mehdipour , J.S. Kaastra , , , and A.J.J. Raassen , SRON Netherlands Institute for Space Research, Sorbonnelaan 2, 3584 CA Utrecht, the Netherlandse-mail:
[email protected] Department of Physics and Astronomy, Universiteit Utrecht, P.O. Box 80000, 3508 TA Utrecht, the Netherlands Leiden Observatory, Leiden University, PO Box 9513, 2300 RA Leiden, the Netherlands Astronomical Institute “Anton Pannekoek”, Science Park 904, 1098 XH Amsterdam, University of Amsterdam, the NetherlandsReceived 15 April 2015 / Accepted 22 May 2015
ABSTRACT
He-like ions produce distinctive series of triplet lines under various astrophysical conditions. However, this emission can be a ff ectedby line absorption from Li-like ions in the same medium. We investigate this absorption of He-like triplets and present the implicationsfor diagnostics of plasmas in photoionisation equilibrium using the line ratios of the triplets. Our computations were carried out forthe O vi and Fe xxiv absorption of the O vii and Fe xxv triplet emission lines, respectively. The fluorescent emission by the Li-likeions and continuum absorption of the He-like ion triplet lines are also investigated. We determine the absorption of the triplet lines asa function of Li-like ion column density and velocity dispersion of the emitting and absorbing medium. We find O vi line absorptioncan significantly alter the O vii triplet line ratios in optically-thin plasmas, by primarily absorbing the intercombination lines, and toa lesser extent, the forbidden line. Because of intrinsic line absorption by O VI inside a photoionised plasma, the predicted ratio offorbidden to intercombination line intensity for the O VII triplet increases from 4 up to an upper limit of 16. This process can explainthe triplet line ratios that are higher than expected and that are seen in some X-ray observations of photoionised plasmas. For theFe xxv triplet, line absorption by Fe xxiv becomes less apparent owing to significant fluorescent emission by Fe xxiv . Without takingthe associated Li-like ion line absorption into account, the density diagnosis of photoionised plasmas using the observed line ratios ofthe He-like ion triplet emission lines can be unreliable, especially for low- Z ions. Key words.
Techniques: spectroscopic – Atomic processes – Atomic data – X-rays: general
1. Introduction
Atomic transitions and their resulting spectral lines provide awealth of useful information about astrophysical plasmas in theuniverse. High-resolution X-ray spectroscopy is key for any di-agnosis of hot plasmas. Over the past 15 years, the grating spec-trometers onboard
Chandra (LETGS, Brinkman et al. 2000 andHETGS, Canizares et al. 2000) and
XMM-Newton (RGS, denHerder et al. 2001) have been instrumental in extracting impor-tant physical information from hot plasmas. One example of im-portant spectral lines is that of He-like ions, which have beenused for density and temperature diagnostics.The first six excited states of a He-like atom haveslightly di ff erent energy levels, depending on the angu-lar momentum and spin of the excited electron, and havefour line transitions to the ground state: the resonanceline w (1 s p P → s S ), the intercombination lines x (1 s p P → s S ) and y (1 s p P → s S ), and the for-bidden line z (1 s s S → s S ). Counting the x and y linesas one line, they are often called the He-like triplet. For the anal-ysis of solar coronae, Gabriel & Jordan (1969) first proposedthat the relative intensities of the lines in a He-like triplet can beused for temperature and density diagnostics. They introducedline ratios, which depend on electron density n e , and temperature T e , which are R ( n e ) = z / ( x + y ) and G ( T e ) = ( z + ( x + y )) / w ,where w , x + y , and z stand for the intensities of the resonance,intercombination and forbidden lines, respectively. Since thenother studies have advanced our understanding of He-like ion diagnostics in collisional ionisation equilibrium (CIE) plasmas,as well as extending the diagnosis to plasmas in photoionisationequilibrium (PIE) and non-equilibrium ionisation (NEI). For ex-ample, Porquet & Dubau (2000) have computed and comparedthe R and G ratios for six He-like ions for plasmas under CIE,PIE, and hybrid conditions. For a review of He-like diagnosticstudies, see Porquet et al. (2010) and references therein.The absorption lines of a Li-like ion occur close to theemission lines of its He-like ion triplet. The absorption linescan therefore a ff ect the intensity of the triplet lines, in partic-ular, those of the intercombination lines as shown in the presentstudy. Without taking the Li-like ion line absorption intrinsic toa medium emitting He-like triplets into account, incorrect tripletline ratios can be obtained, which result in misleading plasmadiagnostics.
2. Diagnostics of photoionised plasmas using X-raydetections of He-like ion triplets
In the literature there are reported X-ray detections and diagnos-tics of He-like triplets in emission from photoionised plasmas inactive galactic nuclei (AGN). Collinge et al. (2001) identified theforbidden and intercombination lines of both O vii and Ne ix inNGC 4051. The R -ratio was found to be about 5 for both triplets.Using this R -ratio and the theoretical calculations of Porquet &Dubau (2000), they obtain an upper limit of 4 × cm − forthe density n e . McKernan et al. (2003) have estimated an upper
1. Mehdipour et al.: Absorption of He-like ion triplets by Li-like ion lines
Table 1.
Atomic parameters of those O vi and Fe xxiv transitions, which contribute to the line absorption of the O vii and Fe xxv triplet lines. Ion λ c (Å) E c (keV) Upper energy level f osc A rad (s − ) A aut (s − ) A tot (s − ) ω O vi . . s s ( S)2 p P / . × − . × . × . × . × − O vi . . s s ( S)2 p P / . × − . × . × . × . × − O vi . . s s ( S)2 p P / . × − . × . × . × . × − O vi . . s s ( S)2 p P / . × − . × . × . × . × − Fe xxiv . . s ( S)2 s p ( P) P / . × − . × . × . × . × − Fe xxiv . . s ( S)2 s p ( P) P / . × − . × . × . × . × − Fe xxiv . . s ( S)2 s p ( P) P / . × − . × . × . × . xxiv . . s ( S)2 s p ( P) P / . × − . × . × . × . × − Fe xxiv . . s ( S)2 s p ( P) P / . × − . × . × . × . × − Notes. λ c and E c are the line-centre wavelength and energy, respectively. A rad , A aut and A tot are the radiative, autoionisation and total transitionprobabilities, respectively. The oscillator strength f osc and fluorescence yield ω are dimensionless. For all lines the lower energy level is 1 s s S / . limit of R < ix and R <
10 for O vii in NGC 4593; theseconstraints imply di ff erent densities of n e < × cm − and n e < × cm − , respectively. Recently, Landt et al. (2015)found that the O vii R -ratio from RGS spectra of NGC 4151 issignificantly higher than the one from their theoretical calcula-tions for a PIE plasma using the Cloudy code (Ferland et al.1998). For three of their observing epochs, in which the flux ofthe O vii lines is best constrained, they measure R = . ± . . ± . . ± .
5, whereas the predicted theoretical upperlimit is R = .
0. They also find that an additional contributionfrom a CIE plasma would not increase the predicted R -ratio highenough to reach the observed R . Finally, the O vii R -ratio fromRGS spectroscopy is observed to be 5 . ± . . ± . Cloudy , Kraemer et al.(2015) report that the absorption-corrected R = . ± .
8, whichis still higher than their predicted R of 3.9 to 4.0. In general, theline ratios from X-ray detections of He-like triplets in AGN indi-cate some degree of deviation from theoretical predications for aPIE plasma, which may be due to Li-like ion line absorption aswe argue in this paper.
3. Computation of the Li-like ion line absorption ofthe He-like ion triplet lines
The absorbed line intensity I ( λ ) is related to the initial in-tensity I ( λ ) before absorption according to I ( λ ) = I ( λ ) T ( λ ),where T ( λ ) is the transmittance of the absorbing mediumas a function of wavelength λ . For the case of a slab con-taining an emitting and absorbing medium, the transmittance T ( λ ) = (1 − e − τ ( λ ) / cos θ ) / ( τ ( λ ) / cos θ ), where τ ( λ ) is the opticaldepth of the medium and θ is the angle between the line ofsight and the normal to the plane of the slab. In this work, weadopt θ =
0. The optical depth τ ( λ ) = τ φ ( λ ), where τ is theoptical depth at the line-centre λ c and φ ( λ ) is the line profile.The optical depth τ = α h λ c f osc N ion / √ π m e σ v , where α isthe fine structure constant, h the Planck constant, f osc the os-cillator strength, N ion the column density of the absorbing ion, m e the electron mass and σ v the velocity dispersion. The lineprofile φ ( λ ) is modelled with a Voigt line profile H ( a , y ), where a = A tot λ/ π b , y = c ( λ − λ c ) / b λ c . Here, b = √ σ v and A tot isthe total transition probability, which is the sum of radiative A rad and autoionisation A rad transition probabilities. The atomic pa- rameters of the O vi and Fe xxiv transitions, which contributeto the line absorption of the O vii and Fe xxv triplet emissionlines, are given in Table 1. The parameters are obtained from theFlexible Atomic Code ( FAC ) (Gu 2008).The emission lines of the He-like ion triplets were mod-elled using a Gaussian profile. The Voigt profile was not re-quired as the Lorentzian contribution was found to be mini-mal for these lines, hence the Gaussian profile provides a de-cent approximation. The line-centre wavelengths of the O vii triplet lines are 21 . w ), 21 . x ), 21 . y )and 22 . z ). The line-centre energies of the Fe xxv tripletlines are 6 . w ), 6 . x ), 6 . y )and 6 . z ). The wavelengths or energies of the He-like triplets and the Li-like absorption lines are from the NISTAtomic Spectra Database v5 and Schmidt et al. (2004). In ourcomputations, σ v for the He-like ions was fixed to that of theLi-like ions as they are taken to be part of the same medium.Following absorption by a Li-like ion, the excited ion de-excites by filling the vacancy in its inner shell with an outer-shellelectron. The released energy emerges either as an Auger elec-tron or a fluorescent photon. As the latter results in e ff ectivelycancelling out the Li-like line absorption, its contribution needsto be taken into account. The fluorescence yield ω is given by ω = A rad / ( A rad + A aut ) and generally increases with the nuclearcharge of the ion, thus the Fe xxiv lines have higher ω than theO vi lines (see Table 1). The observed e ff ect of the fluorescentemission was approximated in our calculations by multiplyingthe τ of each absorption line with (1 − ω ). So at ω = τ remainsunchanged and at ω = τ becomes zero as all the absorbed linephotons are re-emitted as fluorescent photons.Figure 1 shows examples of O vii and Fe xxv triplet emis-sion lines, absorbed by their Li-like ion counterparts. The con-tour diagrams of Fig. 2 show the emerging flux fraction of y and z lines of the He-like triplets after Li-like absorption and fluores-cent emission as a function of N Li-like and σ v . The correspondingcontour plot for the intercombination x line is omitted to savespace, as it looks identical (for O vii ) or similar (for Fe xxv ) tothat of the y line. In Fig. 3 we show the column density limits atwhich continuum absorption and self-absorption of the He-liketriplet lines (described in Sect. 3.1) become critical in CIE andPIE plasmas. Finally, Fig. 4 shows how the unabsorbed R -ratioof the He-like triplets is altered because of O vi and Fe xxiv lineabsorption and fluorescent emission.
2. Mehdipour et al.: Absorption of He-like ion triplets by Li-like ion lines
OVI absorption T r an s m i tt an c e N OVI = 10 cm −2 σ v = 300 km s −1 FeXXIV absorption T r an s m i tt an c e N FeXXIV = 10 cm −2 σ v = 300 km s −1 OVII emission & OVI absorption R e l a t i v e i n t en s i t y σ v = 300 km s −1 N OVI = 0N
OVI = 10 cm −2 w x y z σ v = 300 km s −1 N OVI = 0N
OVI = 10 cm −2 w x y z FeXXV emission & FeXXIV absorption R e l a t i v e i n t en s i t y σ v = 300 km s −1 N FeXXIV = 0N
FeXXIV = 10 cm −2 z y x w σ v = 300 km s −1 N FeXXIV = 0N
FeXXIV = 10 cm −2 z y x w Fig. 1.
Top panels : examples of O vi and Fe xxiv transmittance T ( λ ) with N Li-like = cm − and σ v =
300 km s − . Bottom panels :examples of O vii and Fe xxv triplet emission lines (for a PIE plasma case) absorbed by O vi and Fe xxiv lines, respectively. In allpanels the e ff ect of fluorescent emission is indicated using the dashed curves (excluding fluorescence) compared to the solid curves(including fluorescence). The λ c or E c of the He-like (O vii and Fe xxv ) emission lines are indicated with vertical dotted lines andthose of the Li-like (O vi and Fe xxiv ) absorption lines with vertical dashed lines. OVII intercombination (y) N OVI (cm −2 )100200300400500600 σ v ( k m s − ) . . . . . . . . . . . . . . . . . . FeXXV intercombination (y) N FeXXIV (cm −2 )100200300400500600 σ v ( k m s − ) . . . . . . . . . . . . OVII forbidden (z) N OVI (cm −2 )100200300400500600 σ v ( k m s − ) . . . . . . . . . . . . . FeXXV forbidden (z) N FeXXIV (cm −2 )100200300400500600 σ v ( k m s − ) . . . . . . Fig. 2.
Contour diagrams showing the remaining flux fraction of the intercombination y and forbidden z emission lines of O vii andFe xxv triplets, after intrinsic line absorption by the O vi and Fe xxiv ions (including fluorescent emission) as a function of N Li-like and σ v . The green vertical lines in each panel indicate N crit , which is the critical Li-like ion column density limit corresponding tocontinuum optical depth τ cont =
3. Mehdipour et al.: Absorption of He-like ion triplets by Li-like ion lines
At su ffi ciently high column densities, absorption of line photonsin the continuum becomes important. For CIE and PIE plasmacases, we have computed the critical column density ( N crit ) ofthe Li-like ions for which the continuum optical τ cont = hot (CIE) and pion (PIE)models in the SPEX code (Kaastra et al. 1996) v2.05.04 to cal-culate τ cont for electron temperatures, where the ionic concen-trations of O vi and Fe xxiv peak. For elemental abundancesthe proto-solar values of Lodders et al. (2009) were adopted.For the CIE case, these temperatures are found to be 0.0251keV (2 . × K) for O vi and 1.58 keV (1 . × K)for Fe xxiv . For the PIE plasma case, we adopted the ionisingSpectral Energy Distribution (SED) of NGC 5548 obtained byMehdipour et al. (2015) (the 2013 ‘unobscured’ SED version),which is representative of that of a typical Seyfert-1 AGN. Inthis PIE case, the temperatures at which the Li-like ionic con-centrations peak are found to be 0.00177 keV (2 . × K)for O vi and 0.402 keV (4 . × K) for Fe xxiv , respectively.The N crit limits (corresponding to τ cont = vi N crit is 1 . × cm − forthe CIE and 1 . × cm − for the PIE case. The Fe xxiv N crit is 7 . × cm − for the CIE and 5 . × cm − for the PIEcase.In addition to continuum absorption, self-absorption of theHe-like triplet lines also occurs. While the forbidden and inter-combination line have in general a small optical depth τ forline absorption, the resonance line has a significant τ . This isbecause f osc of the resonance line is much higher than thoseof the intercombination and forbidden lines. For O vii , f osc is0.696 for w , which is higher than f osc of x , y and z by a fac-tor of about 5 . × , 5850 and 2 . × , respectively. ForFe xxv , f osc is 0.798 for w , which is higher than f osc of x , y and z by a factor of about 4 . × , 12 . . × , respec-tively. Thus, after the w line, the intercombination y line has thehighest f osc . The e ff ective optical depth ( τ e ff ) for an absorptionline includes contribution from both τ cont and τ , which is givenby τ e ff ≈ √ τ cont ( τ cont + τ ) (Rybicki & Lightman 1979). Thus,if τ e ff ≥
1, the line has a significant probability of being ab-sorbed on its scattering path through the medium. For the O vii and Fe xxv lines, we calculated τ and τ e ff as a function of N Li-like and σ v . This is done for the above CIE and PIE plasma cases, fortemperatures at which ionic concentrations of O vi and Fe xxiv peak. In Fig. 3, the contours corresponding to τ = τ e ff = w and the intercombination y line of O vii andFe xxv are shown. For the intercombination x and forbidden z line, the τ = τ e ff = ≫ N crit ).
4. Discussion and conclusions
As evident in Figs. 1 and 2, Li-like ion line absorption can signif-icantly diminish the flux of He-like ion triplet lines, in particularthose of O vii . The intercombination ( x and y ) line is most af-fected by this line absorption. For instance, N O vi = cm − ,results in 35–80% (depending on σ v ) of the emitted O vii inter-combination line to be observed (see Fig. 2). The forbidden z lineis much less a ff ected by Li-like absorption than the intercombi-nation line. The line absorption of z is however more dependenton σ v than that of the intercombination line, becoming greater OVII triplet (column density limits) N OVI (cm −2 )100200300400500600 σ v ( k m s − ) τ = ( w ) τ e ff = ( w ) τ c on t = τ = ( y ) τ e ff = ( y ) FeXXV triplet (column density limits) N FeXXIV (cm −2 )100200300400500600 σ v ( k m s − ) τ = ( w ) τ e ff = ( w ) τ c on t = τ = ( y ) τ e ff = ( y ) Fig. 3.
Contour diagrams corresponding to τ = τ e ff = w and intercombination y line of O vii ( top panel ) and Fe xxv ( bottom panel ). The green vertical lines ineach panel indicate N crit , which is the critical Li-like ion columndensity limit corresponding to continuum optical depth τ cont = σ v . The resonance w line remains almost com-pletely una ff ected by the Li-like line absorption (see Fig. 1). The w line is instead susceptible to absorption in the continuum dueto its long path length caused by resonance scattering.The predicted upper limit on the R -ratio for O vii is about 4.0for a PIE plasma according to Cloudy calculations. This upperlimit value decreases with the nuclear charge and reaches ∼ xxv . For the examples shown in Fig. 1, the R -ratio with-out Li-like absorption is 4.0 for O vii and 1.0 for Fe xxv . WithLi-like absorption and the associated fluorescent emission, theemerging R -ratio changes from 4.0 to 9.7 for O vii , and from1.0 to 1.05 for Fe xxv . Importantly, this is a large increase inthe R -ratio for O vii , whereas that of Fe xxv increases by a smallamount as Fe xxiv line absorption of the Fe xxv intercombina-tion line is much less e ff ective than O vi line absorption of theO vii intercombination line (see Fig. 2). This is due to significantfluorescent emission by Fe xxiv , cancelling out the e ff ect of itsline absorption. Without fluorescent emission, the emerging R -ratio after absorption would be 9.5 for O vii and 1.7 for Fe xxv in the examples of Fig. 1. Thus, the e ff ect of fluorescent emis-
4. Mehdipour et al.: Absorption of He-like ion triplets by Li-like ion lines N OVI (cm −2 )100200300400500600 σ v ( k m s − ) OVII (absorbed R / unabsorbed R) . . . . . . . . . . . . . . . . . . . . . . N FeXXIV (cm −2 )100200300400500600 σ v ( k m s − ) FeXXV (absorbed R / unabsorbed R) . . . . . . . . . . . . . Fig. 4.
The absorbed R -ratio divided by the unabsorbed R -ratiofor the O vii ( top panel ) and Fe xxv ( bottom panel ) triplets as afunction of N Li-like and σ v . The contour lines show the factors bywhich the R -ratio value changes due to Li-like line absorption,including the e ff ect of fluorescent emission. The green verticallines in each panel indicate N crit , which is the critical Li-like ioncolumn density limit corresponding to continuum optical depth τ cont = vi absorption of the O vii lines and much moresignificant for Fe xxiv absorption of the Fe xxv lines.Figure 2 shows that below the N crit limit (i.e. in the optically-thin regime of τ cont < vi line absorption of the O vii tripletintercombination line is significant. Thus, the resultant R -ratiowill be significantly altered from its theoretical unabsorbedvalue. Figure 4 shows that O vi line absorption causes the O vii R -ratio to increase by a factor of up to 3.9 at σ v ≈
200 km s − ,hence the predicted upper limit of R ≈ . ≈
16 due to the O vi line absorption. Indeed, thisis consistent with the higher than expected O vii R -ratio val-ues found in AGN as described in Sect. 2 (e.g. Landt et al.2015). Interestingly, Fig. 4 also reveals that at only high σ v ( &
500 km s − ), the R -ratio can actually become smaller in theoptically-thin regime, due to higher line absorption of the for-bidden line as σ v increases. For the Fe xxv triplet, the maximumchange in the R -ratio achieved by Fe xxiv line absorption is onlya factor of 1.3 in the PIE case and 1.4 in the CIE case.We note that in both CIE and PIE plasmas, Li-like ionline absorption and hence R -ratio changes are possible in the optically-thin regime (see Fig. 4). In principle, Li-like ion lineabsorption of the He-like ion triplet lines is intrinsic to any kindof plasma containing these ions. However, for this absorption tobe able to significantly alter the R -ratio of the triplets and be-come observationally distinguishable, su ffi ciently high columndensity in our line of sight is required, such as those found in thePIE plasmas of AGN.The e ff ective optical depth τ e ff of the resonance w line (pre-sented in Fig. 3) shows that τ e ff > N crit ). This means the G -ratio becomes alteredas the w line self-absorption becomes significant in the optically-thin regime. This is the case for both O vii and Fe xxv triplets inboth CIE and PIE plasmas. On the other hand, τ e ff of the in-tercombination and forbidden lines reach unity at much highercolumn densities (above N crit ) due to their low f osc , and thustheir impact on the R -ratio in the optically-thin regime is min-imal. Although in the case of Fe xxv intercombination y line,the column density corresponding to τ e ff = N crit if σ v <
200 km s − . As long as N Li-like is lower than the col-umn density corresponding to τ e ff = N Li-like > N crit , the R -ratioremains almost unchanged. This is because the wavelengths ofthe intercombination and forbidden lines are so close that theyexperience almost the same continuum absorption.We note that in the case of a transient plasma such as insupernova remnants (not investigated in the present study), theinner-shell ionisation of Li-like ions can also significantly in-crease the R -ratio value due to apparent increase in the intensityof the He-like forbidden line, in particular for high- Z ions suchas Fe xxv . Additionally, the R -ratio is also dependent on the tem-perature of the plasma (albeit to lesser extent than the density)and reaches higher values in the case of a CIE plasma.In this paper we use a simple slab geometry to demonstratethe principles of intrinsic Li-like line absorption and its e ff ectson the He-like triplet emission lines without introducing addi-tional complexities. We note that while for resonant scattering(hence changes in the G -ratio) the details of the plasma geometrybecome important, Li-like line absorption (and hence changesin the R -ratio) is primarily related to the optical depth of themedium in the line of sight, which is dependent on the columndensity and dispersion velocity as presented in this paper.The future high-resolution X-ray spectrometers of the Astro-H and
Athena observatories will be useful for precise parameter-isation of the He-like triplets lines. The measurements of the lineparameters and flux using the existing grating spectrometers of-ten give large uncertainties on the R -ratio due to low count-ratestatistics. At present there are only very limited accurate mea-surements of the R -ratio from He-like triplets in the X-ray band(mostly O vii ) for photoionised plasmas. The microcalorimeterspectrometers of Astro-H and
Athena will combine high sensi-tivity with unprecedented spectral resolution at the 6 keV band.They will enable us to detect not only the He-like triplets in thesoft X-ray band, but also the highly ionised Fe xxv triplet at hardX-rays. This triplet is key for plasma diagnostics at the high-temperature and high-density domain. Using these upcoming in-struments, the R -ratio and the associated Li-like line absorptioncan be investigated for di ff erent He-like triplets in a decent sizesample of objects.In conclusion, intrinsic line absorption by Li-like ions in aphotoionised medium can significantly diminish the intensity ofthe intercombination line of its He-like ion triplet emitted insidethe same medium. As a result, this absorption causes significantalteration in the line ratios of the triplets. This is the case for O vi column densities between 10 and 10 cm − in the optically-
5. Mehdipour et al.: Absorption of He-like ion triplets by Li-like ion lines thin regime. The predicted R -ratio for O vii , increases from 4to an upper limit of 16 due to this process. For Li-like ions withhigher nuclear charge (such as Fe xxiv ), the e ff ect of line absorp-tion becomes less apparent due to strong fluorescent emissionby these ions. Finally, we emphasise that without consideringthe line absorption by Li-like ions, the use of observed He-liketriplet line ratios can give erroneous density diagnostics for pho-toionised plasmas. Acknowledgements.
SRON is supported financially by NWO, the NetherlandsOrganization for Scientific Research. We thank the anonymous referee for her / hisuseful comments. References