An Updated Formalism For Line-Driven Radiative Acceleration and Implications for Stellar Mass Loss
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An Updated Formalism For Line-Driven Radiative Acceleration and Implications for Stellar MassLoss
Aylecia S. Lattimer and Steven R. Cranmer Department of Astrophysical and Planetary Sciences, Laboratory for Atmospheric and Space Physics, University of Colorado, Boulder,CO 80309, USA (Accepted January 23, 2021)
ABSTRACTRadiation contributes to the acceleration of large-scale flows in various astrophysical environmentsbecause of the strong opacity in spectral lines. Quantification of the associated force is crucial tounderstanding these line-driven flows, and a large number of lines (due to the full set of elements andionization stages) must be taken into account. Here we provide new calculations of the dimensionlessline strengths and associated opacity-dependent force multipliers for an updated list of approximately4.5 million spectral lines compiled from the NIST, CHIANTI, CMFGEN, and TOPbase databases. Tomaintain generality of application to different environments, we assume local thermodynamic equilib-rium, illumination by a Planck function, and the Sobolev approximation. We compute the line forcesin a two-dimensional grid of temperatures (i.e., values between 5,200 and 70,000 K) and densities(varying over 11 orders of magnitude). Historically, the force multiplier function has been describedby a power-law function of optical depth. We revisit this assumption by fitting alternative functionsthat include a saturation to a constant value (Gayley’s ¯ Q parameter) in the optically-thin limit. Thisalternate form is a better fit than the power-law form, and we use it to calculate example mass-lossrates for massive main-sequence stars. Because the power-law force multiplier does not continue toarbitrarily small optical depths, we find a sharp decrease, or quenching, of line-driven winds for starswith effective temperatures less than about 15,000 K. Keywords:
Stellar winds (1636), Stellar mass loss (1613), Atomic Physics (2063), Radiative Processes(2055) INTRODUCTIONThe study of flows driven by photons has a long his-tory. For example, Johnson (1925) hypothesized thatthat helium atoms could be ejected from a star via radi-ation pressure, suggesting that photon-driven flows canbe responsible for stellar mass loss. We now know thatphoton-driven outflows exist in various astrophysical en-vironments, including accreting objects such as proto-stars, cataclysmic variables, and active galactic nuclei(AGN) (Owocki 2004; Puls et al. 2008; Higginbottomet al. 2014). The driving mechanism of these flowsby photons is often referred to as “radiation pressure,”where the force of radiation from the spectral lines actson the material of the outflow. The very large number ofspectral lines in any given ion has a dominant effect on
Corresponding author: Aylecia S. Lattimer this pressure on the flow material (Castor 1974; Castoret al. 1975). The absorption and re-emission of pho-tons in a spectral line of frequency ν results in a radialtransfer of momentum, driving a wind outward from thestar.Quantifying and understanding mass and energy flowsis critical in the understanding of how affected objectsevolve and interact with their surroundings. Massivestar winds can be directly observed in the stellar spec-tral energy distributions once the star is above a certainthreshold luminosity (Kudritzki & Puls 2000). Ultravio-let (UV) observations showing P Cygni profiles and highflow velocities initially suggested that an extension of so-lar wind theory was insufficient to explain the winds ofO and B stars (Cassinelli 1979). Lucy & Solomon (1970)demonstrated that the absorption and re-emission ofphotons by 12 UV resonance lines can drive a wind in Otype stars, consequently driving mass loss in the form of a r X i v : . [ a s t r o - ph . S R ] J a n the stellar wind. Castor et al. (1975) (hereafter CAK)found that the large number of lines in any given ionof an atom contribute to the radiation force, propor-tional to the continuum flux at their specific frequency.Using 900 multiplets of C III, CAK found a mass lossrate estimate ∼
100 times greater than those predictedby Lucy & Solomon (1970), leading to the conclusionthat the force from the lines should exceed gravity byapproximately two orders of magnitude for small opti-cal depths in O type stars. Therefore, O stars cannothave static atmospheres, as there is no mechanism thatcan prevent the ejection of the surface layers of the star.This was a major advancement in the understanding ofstellar winds. Models based on the theory of line-drivenwinds have yielded results that agree well with observa-tions of mass loss and terminal flow rates (Owocki et al.1988; Puls et al. 2008; Sundqvist et al. 2019).Line-driven outflows are also encountered in variousother environments. For example, broad absorptionlines are present in the UV spectra of quasars, as wellas in other wavelengths. In these cases, the blueshift ofthe lines suggests the presence of winds from the AGN,sometimes with velocities of up to 0 . c , making a line-driven disk wind a promising hydrodynamical scenariofor AGN outflows (Proga 2007; Risaliti & Elvis 2010).Additionally, Kee et al. (2016) suggested that the strongline-driven winds from OB type stars with circumstellardisks could drive ablation of the disk’s surface layers onshort timescales that could be a contributing factor tothe relative rarity of O type stars in the galaxy (see alsoKee et al. 2018a, Kee et al. 2018b, Kee & Kuiper 2019).By themselves, winds from massive stars can drive galac-tic evolution by injecting momentum, energy, and stellarmaterial into the interstellar medium (Kudritzki & Puls2000).Previous authors have computed lists of spectral linesand modeled the distributions of the line strength pa-rameters (see for example Abbott 1982; Shimada et al.1994; Gormaz-Matamala et al. 2019). However, theseline lists have been comprised of significantly fewer linesthan those currently available. Lucy & Solomon (1970)originally used 12 lines, CAK updated this to include900 transitions, and Pauldrach (1987) used a list of10,000 transitions. More recently, Gormaz-Matamalaet al. (2019) used a list of ∼ M ( t ), as well as the fittingof a CAK-form power-law (Section 4.2) and an alternatefunction (Section 4.3) to the calculated values. Section5 describes the calculation of mass-loss rates from boththe CAK (Section 5.2) and alternate (Section 5.3) mul-tiplier forms, and a comparison of the two (Section 5.5).We end with a discussion of our conclusions in Section6. RADIATIVE ACCELERATION OFLINE-DRIVEN FLOWSA general way of expressing the radiative acceleration g rad on a parcel of gas is to take the first moment ofthe equation of radiative transfer. Following Hubeny &Mihalas (2014), g rad ( r ) = 1 c (cid:90) dν (cid:90) d Ωˆ n ( κ ν I ν − j ν ) (1)where ˆ n is the unit vector specifying an arbitrary ray-path, κ ν and j ν are the absorption coefficient (cm g − )and emissivity, and I ν is the specific intensity. In thecomoving frame of a flow, it is often assumed that j ν is an even function of ˆ n (so it cancels out of the abovemoment integral) and that angle anisotropies of κ ν aresufficiently weak to allow it to be taken out of the solid-angle integral. Thus, the expression we use is g rad ( r ) = 1 c (cid:90) κ ν F ν ( r ) dν (2)where F ν is the radiative flux (photon energy flux). Be-low, we also write the opacity χ ν in units of cm − as χ ν = κ ν ρ = σ ν n (3)where it is sometimes useful to use the absorption coef-ficient κ ν or the cross section σ ν , and we also define themass density ρ and number density n in units of cm − .From Equation (2), we see that opacity is necessary forany acceleration due to the radiation field to occur; with ine Driven Winds Dimensionless Line Strengths
We follow Gayley (1995) in characterizing the distri-bution of spectral line strengths as a set of dimensionlessratios q i that describe the atomic physics, and dimen-sionless weighting factors (cid:102) W i that describe the illumi-nation of the atom from a given spectral energy dis-tribution. The product q i (cid:102) W i represents the full ratioof radiative acceleration due to a specific line i to theacceleration on free electrons. In such a ratio of acceler-ations (see Equation (2)), the pre-factors of 1 /c cancelout, and we choose to multiply both the numerator anddenominator by the mass density ρ . We can write thebound line opacity as χ ν = χ L φ ( ν ) (4)where φ ( ν ) is the line profile function. Thus, the ratioof accelerations can be written as g bound g free = (cid:82) χ ν F ν dν (cid:82) χ e F ν dν = χ L (cid:82) φ ( ν ) F ν dνχ e (cid:82) F ν dν . (5)For this work, we assume that the environments in ques-tion are in LTE. This assumption (also used by CAK)is often set aside when modeling line-driven winds from,e.g., massive stars, but here we retain it to maintain alevel of generality concerning the environments in ques-tion. Of course, this assumption will need to be reevalu-ated in future applications to specific systems (see, e.g,Pauldrach 1987). Under the assumption of LTE, thequantity χ L is then given by χ L = πe m e c f ij n i (cid:16) − e − hν /kT (cid:17) , (6)where f ij is the semiclassical oscillator strength, where i and j are used to refer to the lower and upper atomiclevels, respectively. The Thomson scattering opacity χ e is given by χ e = σ T n e = (cid:18) πr e (cid:19) n e (7) where σ T is the Thomson scattering cross section andthe classical electron radius is given by r e = e m e c . (8)Each line profile function φν ) is very narrow whenintegrated over the energy distribution, so we can ap-proximate it to a Dirac delta function when evaluatedat frequency ν (cid:90) φ ( ν ) F ν dν ≈ F ν . (9)We can additionally define the frequency integrated flux F as (cid:90) F ν dν ≡ F. (10)We can then define a dimensionless weighting ratio (cid:102) W i = ν F ν F , (11)which accounts for the flux integrals in Equation (3).In this paper, we maintain a generality of applicationby assuming a Planck function for the illuminating flux.We use a temperature T , presumed equal to the localtemperature of the gas ( T = T rad = T eff ), to characterizethis Planck function (Noebauer & Sim 2015; Gormaz-Matamala et al. 2019). Although some authors havesuggested the electron temperature is a fraction (usu-ally 0.8 T eff ) of the effective temperature, the radiativetemperature is often taken to be equal to that of T eff inthe Planck case (see, for example, Puls et al. 2000). Thisassumption is commonly used for the wind-driving cir-cumstellar regions near massive-star photospheres (e.g.,Drew 1989, Kee et al. 2016), in which radiative cool-ing in an isothermal gas efficiently maintains a nearlyconstant temperature T , roughly equal to the stellar ef-fective temperature T eff . Thus, we specify F ν = 2 πhν /c e hν /k B T − , (12)and the frequency-integrated flux is given by F = σT , (13)where σ is the Stefan-Boltzmann constant. Since theseare dependent on the wavelength of the transition, thereis a unique weighting parameter (cid:102) W i for each line in everyion and temperature T .Using Equation (11), we can write the ratio of thebound to free acceleration as g bound g free = χ L χ e (cid:102) W i ν = χ L χ e λ (cid:102) W i c . (14)Finally, combining Equations (6), (7), and (14), wehave g bound g free = 38 λ r e f ij n i n e (cid:16) − e − hν /k B T (cid:17) (cid:102) W i , (15)and we define the dimensionless line strength parameter q i as q i ≡ λ r e f ij n i n e (cid:16) − e − hν /k B T (cid:17) . (16)This is similar to Equation 9 of Gayley (1995). However,here we have included the traditional correction factorfor stimulated emission.We also define the sum of the line strengths ¯ Q as¯ Q = (cid:88) i q i (cid:102) W i . (17)2.2. Ionization Equilibrium
In order to calculate Equation (16), we first must firstcalculate the number density ratio n i /n e , which is givenby n i n e = (cid:18) n i n ion (cid:19) (cid:18) n ion n el (cid:19) (cid:18) n el n H (cid:19) (cid:18) n H n e (cid:19) . (18)In LTE, the total number of particles in the lower tran-sition level can be described as a fraction of the totalnumber of particles in the given ionization state: n i n ion = g i e − ( E i − E ) /k B T U ion ( T ) . (19)Here U ion is the ion-specific temperature dependent par-tition function, g i is the ground state statistical weight, E i is the energy of the lower level of the transition, and E is the ground level energy, which was set to zero bythe atomic databases used here. It should also be notedthat the oscillator strength f ij (as in Equation (16)) isneeded only to form the product g i f ij (as in Equation(19)) and never appears alone.The elemental abundance ratio n el /n H was obtainedfrom tabulated solar abundances (Asplund et al. 2009),whereas the other two quantities in parentheses, n ion /n el and n H /n e , were found using the Saha equation: n i +1 n i n e = 2 λ e U i +1 U i exp (cid:20) − E i +1 − E i k B T (cid:21) , (20)where the thermal de Broglie wavelength of a free elec-tron λ e is given by λ e = h/ (cid:112) πm e k B T . (21)To solve Equation (20) for the pairwise ionization frac-tions n i +1 /n i , we also need to know the electron num-ber density n e . Therefore, we made an initial estimate log T (K) n e m H / Figure 1.
Iterated values of n e as a function of temperature,for ρ = 10 − g cm − . for n e , which was then refined using an undercorrectiontechnique. This was done at the end of each iterationover the Saha equation by tabulating a new estimateof n e from the calculated ionization balance, which wasthen multiplied by the previous estimate. The squareroot of this product was then used as the estimate of n e for the next iteration. We found that 20 iterations weresufficient to reach a stable value for n e . An example setof calculations is shown in Figure 1, which shows thefinal converged-upon values of n e for our temperaturerange, for an example density of ρ = 10 − g cm − .The initial estimate for n e was given by n e = 0 . (cid:18) ρm H (cid:19) (22)where ρ is the density of the wind and m H is the massof Hydrogen. For this work, we used a grid of densitywith 22 values spanning 10 − to 10 − g cm − , a similardensity range to that used by Abbott (1982). This broadrange is applicable to both massive star winds as wellas other astrophysical environments that exhibit line-driven outflows.The quantity n el can be found from the initial elemen-tal abundances, here taken from Asplund et al. (2009).These are given in the form of number density ratios tothe abundance of hydrogen (i.e. n el /n H ). From these wefind the fractional abundance by mass µ of each element: µ = A el ( n el /n H ) (cid:80) el A el ( n el /n H ) (23)where A el is the atomic weight of the element. The num-ber density in cm − for each element can then be found ine Driven Winds ρ , the fractional abundance µ , and the mass of hydrogen m H : n el = ρµA el m H . (24)Knowing n e and the tabulated number density of hy-drogen n H from Equations (23)–(24), we can then calcu-late the quantity n H /n e , leaving only the second quan-tity in Equation (18), n ion /n el . For ionization stage Ithis is given by n I n el = (cid:20) n II n I + n III n II n II n I + ... (cid:21) − , (25)where II and III represent ionization stages I and II ofthe element in question. Here n I represents n ion as seenin Equation (18). Each fraction in the brackets is givenby the Saha equation (Equation (20)). We can simi-larly isolate n II /n el and the higher ratios to find n ion /n el for any ionization stage, and consequently n ion /n e and n i /n e as in Equation (18), of any element that we con-sider. Figure 2 shows an example calculation of n ion /n el for the ionization stages of oxygen. These steps werecarried out for all elements from H to Ni. We consid-ered all ionization stages for elements with atomic num-bers Z ≤
10, and only the first ten ionization stages forelements with
Z > log T (K) n i o n / n e l I II III IV V VI VII VIII IX
Figure 2.
The ionization fraction n ion /n el for the ioniza-tion stages of oxygen, for a fixed wind density of ρ = 10 − g cm − . 3. ATOMIC DATA3.1.
Partition Functions
The LTE partition functions used in Equations (19)–(20) were calculated according to the fitting procedure set forth by Cardona et al. (2010). Namely, for elementswith Z ≤
20, we used the tabulated fit parameters in theexpression U = g jk + G jk e − (cid:15) jk /k B T + m n ∗ − e − E n ∗ jk /k B T (26)where g jk is the ground state statistical weight, and E n ∗ jk is given by E n ∗ jk = χ jk − Z Ry n ∗ (27)for ionization state j of element k . Here χ jk is the ion-ization potential, Ry is the energy of one Rydberg (13.6eV), and Z eff is the effective ion charge j + 1. The effec-tive maximum upper level index n ∗ is given by n ∗ = q (cid:18) (cid:114) q (cid:19) , with q = (cid:114) Z eff πa n − / (28)where a is the Bohr radius. The total number densityof the gas n tot is given by n tot = n e + (cid:88) n el , (29)with n e given by the iterative undercorrection processdescribed above, and n el given by Equation (24).The quantities m , G jk and (cid:15) jk are drawn from Ta-ble 1 of Cardona et al. (2010). The values of g jk and χ jk are taken from CHIANTI for all modeled elementsand ionization stages (see Section 3.2). Cardona et al.(2010) does not provide the fit parameters for the parti-tion function for elements of Z >
20, and a simple fittingprocedure on the available parameters was performed toempirically calculate estimates of these parameters forhigher- Z elements. The Cardona (cid:15) jk parameter corre-lated fairly well with ionization potential with no morethan ∼
20% error, while the m and G parameters arewell correlated with each other and weakly correlatedwith the ground state statistical weight g jk . These fitsare given by (cid:15) jk = χ jk (0 . − . Z eff ) , (30) m = 4( g jk ) . , (31)and G jk = 113 m . . (32)These were used for all considered elements with Z >
Z > U ≈ g jk . Doing this for the grids of tem-perature and density discussed below resulted in onlynegligible differences in the values of U . 3.2. Database Selection
To find the total radiative force on a parcel of gas fora given temperature and density, we need to constructthe most complete line list of atomic data possible, asall spectral lines encountered by the radiation field mustbe accounted for. To this end, we compiled spectral linedata from four sources: the National Institute of Stan-dards and Technology (NIST) (Kramida et al. 2018),version 9.0 of the CHIANTI database (Dere et al. 1997,2019), the database of lines used by the radiative trans-fer code CMFGEN (Hillier 1990; Hillier & Miller 1998;Hillier & Lanz 2001), and the Opacity Project’s TOP-base (Cunto & Mendoza 1992; Cunto et al. 1993). Theuse of multiple databases was necessary, as there existgaps in the atomic data available from each individualdatabase.We retrieved energy level classifications and wave-length data for each ion. For each selected element andionization stage, tabulations of line oscillator strengths(i.e., g i f ij ) were extracted, along with lower-level ener-gies E i and rest-frame wavelengths λ . These are theparameters necessary to compute the line strength pa-rameter q i . We did this for all elements up to Ni. Forthe ionization stages, we retrieved data for each elementup to nine times ionized (that is, data were retrievedfor ionization stages I through X). For elements withatomic number Z <
10, we retrieved data for all theavailable ionization states, up to fully ionized. In Ap-pendix A, we summarize the process used to determinewhich database would be used for each ion. The finalline list contains 4,514,900 lines. The detailed break-down of line counts and the database used for each ionare given in Table A1.Figure 3(a) shows histograms for the occurrence fre-quency of different values of q i (cid:102) W i for one examplechoice of the local temperature and density. All plottedhistograms were constructed using 95 logarithmicallyspaced bins. Stacked underneath the uppermost curve(corresponding to the total counts in each bin) are indi-vidual histograms that break out the contributions fromeach individual element. In Figure 3(b) we also show his-tograms computed from a subset of the CMFGEN andCHIANTI databases that includes only lines that havebeen observed experimentally (i.e., 589,186 lines out ofthe full set of 4,514,900 lines). Thus, this panel ignoresthe vast majority of lines (i.e., 87%) with only theoreti-cally predicted properties. There is a notable drop-off in Atomic data used here are those which were updated byD.J. Hillier in 2016 (http://kookaburra.phyast.pitt.edu/hillier/cmfgen files/atomic data 15nov16.tar.gz). line strengths below q i (cid:102) W i ≈ − for the distributionthat excludes the theoretical lines. While the contribu-tions by low- Z elements are the same in both cases, thisdrop-off in line strengths represents a lack of observedlines for high- Z elements, most notably cobalt. How-ever, in both cases there is a significant contributionto the distribution by high- Z elements, notably that ofiron, at high line strengths. For the sake of complete-ness, we use the line list comprised of both theoreticaland observed transitions for the remainder of this work.However, for the purpose of comparison Section 4 in-cludes calculations of the force multiplier for both thefull line list and the observed-only subset. N ( q i W i ) HHe CCNOF P FeCo (a)10 q i W i N ( q i W i ) HHe CCOF Ar Fe (b) A t o m i c N u m b e r Z Figure 3.
Comparison of the contributions by element tothe total distribution of line strengths. (a) shows all tran-sitions, including theoretical. (b) excludes theoretical tran-sitions from the CMFGEN and CHIANTI databases. Ma-jor element contributions labeled, for example temperature T = 5200 K and density ρ = 10 − g cm − . In (b) greyindicates contour of the total histogram from (a).4. CALCULATING AND FITTING THE LINEFORCE MULTIPLIERIn order to examine the dynamics of the outflows, wefirst calculate the line force multiplier, a measure of the ine Driven Winds q i (cid:102) W i . We then fit andcompare two functions to the resulting distributions.4.1. Calculating M ( t )The line acceleration is defined as the radiative ac-celeration due to electron scattering, multiplied by theline force multiplier M ( t ). Here, t is an optical depthparameter that is independent of the line strength. It isgiven by t = κ e ρv th (cid:12)(cid:12)(cid:12)(cid:12) dvdr (cid:12)(cid:12)(cid:12)(cid:12) − (33)for expanding atmospheres (Sobolev 1957, 1960, CAK).In the case of a static atmosphere, t is equivalent to theelectron scattering optical depth, while in the expandingcase t is less than this depth.In an expanding wind, we cannot simply take the sumof the q i (cid:102) W i = g bound /g free as found in Equation (15)above. The full force multiplier depends on other ra-diative transfer effects, such as the “self-shadowing” ofthe lines, due to differences in the Doppler-shifted localreference frames (Gayley 1995). Therefore the full cal-culation of the force multiplier M ( t ) from the line listcan be written as M ( t ) = η (cid:88) i q i (cid:102) W i (cid:18) − e − τ i τ i (cid:19) (34)where the geometrical finite-disk factor η is the ratio ofthe true line force to that derived in the limit of purelyradial photons (e.g., it is the same as the ratio F given byEquation (21) of Gayley 1995). Equation (34) assumesthe Sobolev approximation and no overlapping lines, forsupersonic flows. The dimensionless optical depth τ i ofeach line is defined as τ i = cv th q i t (35)where v = 2 k B T /m p is the proton thermal speed forall ions of a single temperature. M ( t ) corresponds to the sum over the spectral linesthat contribute to the wind. Originally, M ( t ) was pa-rameterized by Castor (1974) in terms of the opticaldepth, depending only on the structure of the wind.However, CAK proposed that M ( t ) could take the formof a power-law, expressed as M ( t ) = η kt − α (36)where k and α are the fit constants of the power-law. Be-cause the finite disk factor η appears in both Equations(34) and (36), we can safely neglect it when performingfits and parameterizations for quantities such as k and α . We re-examine the assumption presented in Equa-tion (36) that the line force multiplier takes the form ofa power law. To do this, we will fit both power lawsand an alternative fitting function to the values of M ( t )calculated from the updated line list.Although the CAK t parameter can be evaluated atany point in a radiatively-driven outflow, here we eval-uate M ( t ) over a logarithmic grid of t spanning from10 − to 10. In the limit of t → M ( t ) should become equal to ¯ Q . It should be notedthat Abbott (1982) considers only values of t from 10 − to t = 0 .
1. However, we include values beyond this rangein order to better examine the behavior of M ( t ). For ex-ample, the asymptotic behavior of M ( t ) in the limit ofsmall t is not seen in the range used by Abbott (1982).In calculating the dimensionless line strength parame-ter q i , the weighting function (cid:102) W i , and subsequently theforce multiplier M ( t ) as described above, we used a gridof 100 logarithmically spaced temperatures ranging from5,200 to 70,000 K and 22 densities ranging from 10 − to10 − g cm − . We calculated the force multiplier M ( t )for two cases: (1) the full line list that includes bothobserved and theoretical lines, and (2) only the subsetof observed lines with laboratory wavelengths. Figure 4shows results for M ( t ) for both cases. In general, notethat both higher temperatures and higher densities tendto result in higher values of the force multiplier. Notealso that the turnover to a constant value of M ( t ) ≈ ¯ Q occurs at different values of t ; this is sometimes as low as t ≈ − and sometimes as high as t ≈ − . Stevens& Kallman (1990) calculated the radiative force due toX-ray ionization on the stellar wind in massive X-ray bi-naries. The flattening of the M ( t ) curve seen in Figure 4is similar to Figure 1 of that paper, where the force mul-tiplier is suppressed with increasing X-ray ionization.Figure 5 shows the ratio of the full and observed-onlycalculations for an example value of t ≈
1. The discrep-ancy between the two multipliers is most pronouncedat high temperatures, with the full line list producing amuch higher force multiplier than the observed-only list.This is to be expected, as the observed-only list does notinclude many lines from high- Z elements which requirehigh temperatures to ionize. However, at lower temper-atures the discrepancy between the two calculations isslight. In light of these factors, going forward we useonly the full line list that includes theoretical lines.Figure 6 shows the sum ¯ Q = (cid:80) i q i (cid:102) W i , which was cal-culated for each temperature and density. These arecompared to values from Gayley (1995), who also pro-vided values of ¯ Q as converted from previous works suchas Abbott (1982). Note that temperatures correspond-ing to O and early-B spectral types (log T between 4.2 M ( t ) (a)All Lines (b)Observed Only l o g T ( K ) log t M ( t ) (c) 15.0 12.5 10.0 7.5 5.0 2.5 0.0 log t (d) l o g ( g c m ) Figure 4.
The force multiplier M ( t ): (a) Varying with temperature from T =5,200 K to T =70,000 K for a constant densityof ρ = 10 − g cm − , for observed and theoretical transitions. (b) Varying with temperature from T =5,200 K to T =70,000 Kfor a constant density of ρ = 10 − g cm − , for only observed transitions. (c) Varying with density from ρ = 10 − g cm − to ρ = 10 − g cm − for a constant temperature of T = 5 ,
200 K, for theoretical and observed transitions. (d) Varying withdensity from ρ = 10 − g cm − to ρ = 10 − g cm − for a constant temperature of T = 5 ,
200 K, for only observed transitions. and 4.6) tend to exhibit values of ¯ Q around 10 , indepen-dent of density, as also found by Gayley (1995). Aboveand below this range, there is a strong dependence of ¯ Q on density, with ¯ Q varying drastically at both very lowtemperatures (log T (cid:46) . T (cid:38) . CAK Power Law Fitting
We begin by fitting a power law of the form proposedby CAK (Equation (36)) to the calculated values of M ( t ). This fitting was performed using the Levenberg-Marquardt method of least-squares fitting. As is readilyapparent from Figure 4, the full range of calculated M ( t )values cannot be described by a single power law. There-fore, we fit the initial CAK power law form to only to values of log( t ) > −
3, in order to exclude the flat portionof the curve.As with ¯ Q , we also compare the fitted values of α and k to those from previous work. Figure 7 shows a com-parison to Abbott (1982), Shimada et al. (1994), Gay-ley (1995), and Gormaz-Matamala et al. (2019), with agood agreement with those values, most especially fordensities that fall in the middle of our considered den-sity range. Note that for the lowest values of T , ourvalues for the power-law slope α never get as small assome of the plotted literature values that are of order ∼ M ( t ). For example, Abbott (1982) calculatedvalues of α and k for a range of − ≤ log( t ) ≥ − ine Driven Winds log T (K) l o g ( g c m ) M o b s / M a ll Figure 5.
Ratio of the calculated force multiplier for linelists comprised of all lines to observed lines only, shown herefor an example value of t ≈ log T (K) Q Gayley (1995)Puls (2000) l o g ( g c m ) Figure 6.
Evolution of ¯ Q for our chosen density and tem-perature range, compared to values from Table 1 of Gayley(1995) (black crosses) and Table 2 of Puls et al. (2000) (blacktriangles). temperatures, the flat portions of the M ( t ) curve beginat values of t as high as log( t ) ∼ − .
5. Inclusion of theseflat portions during fitting would yield shallower slopesthan those found in this work.Despite acceptable agreement with previous works forthese two parameters ( α , k ), this preliminary power lawform of the force multiplier presents a decent fit to thecalculated values of M ( t ) for only a narrow range of t ,namely between log( t ) ≈ − t ) ≈ Alternate Fitting Function
We present an alternative fitting function in the formof a saturated power law, given by M ( t ) = η ¯ Qk ( k s + ¯ Q s t αs ) /s . (37) C A K (a) log T (K) k C A K (b) Abbott 1982Gormaz-Matalmala 2019Gayley 1995Shimada 1994Puls 2000 l o g ( g c m ) Figure 7.
Comparison of CAK power law parameters α (a) and k (b) with values from Abbott (1982), Shimadaet al. (1994), Gayley (1995), Puls et al. (2000), and Gormaz-Matamala et al. (2019), for the our chosen range of densities. In this case, α , k , and s are fit parameters, and ¯ Q is thecalculated value as found from the line list and shownin Figure 6. The parameter s is a sharpness parame-ter that determines how rapidly the function transitionsfrom the low- t to high- t limits. This function reduces tothe CAK power law form in the limit of large t , and inthe limit of small t reduces to ¯ Q , consistent with the be-havior of the calculated values of M ( t ). Figure 8 showsthe resulting fits compared to the calculated M ( t ) val-ues, for an example density of ρ = 10 − g cm − . Theabove function was found to be flexible enough to fit thecalculated M ( t ) values quite accurately. For the full gridof parameters ( T , ρ , and t ) we computed the fractionaldifference D between the numerical and best-fit values of M ( t ). The median of this distribution was (cid:104) D (cid:105) = 1 . D > t (cid:38) −
4. How-ever, for values of log t (cid:46) −
4, the fit achieved using thealternate function is drastically improved over that of0the CAK power law. As can be seen in Figure 4, thevalue of t at which the CAK power law begins to fail alsostrongly depends on temperature and density. Addition-ally, we do not include a high opacity cut-off in our newform of M ( t ), such as that suggested by Gayley (1995).Upon calculation of the force multiplier out to high t ( t ∼ ), we find that M ( t ) continues to decrease as apower-law rather than as an exponential drop-off at highopacities, generally approaching the form M ( t ) ∝ t − astemperature and density increase. Therefore, we do notimpose a cut-off of the force multiplier at high opacities.Figure 9 shows the dependence of the saturated powerlaw fit parameters α , k , and s on temperature and den-sity. Also shown is the evolution with temperature anddensity of M ( t ) at t = 1. At high values of t such as t = 1, the force multiplier behaves as a power-law. Thevalues of M ( t = 1) provide an alternate estimation ofthe fit parameter k . Comparison of Figures 9(a) and9(b) additionally provide insight into the deviation ofthe fitted saturated power-law (a) from the actual cal-culated value (b). We also include an estimate of theCAK critical point density across the range of effectivetemperatures (see Section 5.4). Additionally, we indi-cate a locus of parameters at which the Saha ionizationbalance produces equal amounts of Fe III and Fe IV.The relevance of this to a proposed explanation for thephenomenon of “bistability” in line-driven winds is dis-cussed further in Section 6. MASS-LOSS ESTIMATES FOR MASSIVE STARSIn addition to determining a function to better de-scribe the line force multiplier M ( t ), we also explorethe consequences of this updated form on the calculatedmass-loss rates of massive stars. We do this by calcu-lating mass-loss rates for both the traditional power-lawform and our newly updated alternate form.5.1. General Mass-Loss Solution
We begin with the time-steady radial component ofthe momentum conservation equation. To approximatethe supersonic winds of OB stars, we omit the gas-pressure gradient term. Because gas pressure does notplay a fundamental role in highly supersonic winds suchas the ones considered in this work, we can safely ne-glect these terms (see, e.g., Gayley 2000; Owocki 2004).Doing this, we have v ∂v∂r = − GM ∗ r + g free + g bound . (38)Here, v ( r ) is the radially dependent wind velocity, M ∗ is the mass of the central star, and G is the gravita-tional constant. The free radiative acceleration due to M ( t ) (a)14 12 10 8 6 4 2 0 log t M ( t ) (b) 3.84.04.24.44.64.8 l o g T ( K ) Figure 8.
Comparison of alternate fits and calculated val-ues of M(t) for an example density of ρ = 10 − g cm − .(a) M ( t ) as calculated from line list. (b) Fits produced byEquation (37). Thomson scattering g free can be written as the Edding-ton factor Γ times the gravitational acceleration. Withstellar bolometric luminosity L ∗ , the Eddington factorΓ can be written as Γ = κ e L ∗ πcGM ∗ (39)with the mixture-dependent Thomson scattering coeffi-cient given by κ e = σ T n e ρ ≈ σ T m H (cid:18) X (cid:19) (40)where X is the hydrogen mass fraction. The final ap-proximation above is provided only for reference in thelimit of full ionization (see, e.g., Mihalas 1978). In allresults shown below, we use the self-consistent values of n e computed from the Saha equation. For the values of X and Y , we use the bulk composition chemical abun-dances of H and He given in Table 4 of Asplund et al.(2009), with X = 0 . Y = 0 . v ∂v∂r = GM ∗ r [ − M ( t )] . (41) ine Driven Winds l o g ( g c m ) (a) (b) T (K)201816141210 l o g ( g c m ) (c) T (K) (d) l o g ( k ) l o g ( M t = ) l o g ( s ) Figure 9.
Two-dimensional contour plots of the temperature and density dependence of the fit parameters for the alternatefitting function. (a) k , (b) M ( t = 1), (c) α , (d) s . Solid lines indicate estimates of the density at the CAK critical point(Equation (56)), and dashed lines indicate the calculated recombination temperature of Fe between Fe III and Fe IV. As in Gayley (1995), we also define the dimensionlesswind acceleration factor w : w = r vGM ∗ (1 − Γ) dvdr . (42)This allows us to write Equation (41) as F = w + 1 − Γ1 − Γ M ( t ) = 0 . (43)We use this form because the CAK critical-point solu-tion requires at least two conditions to be true for atime-steady wind: F = 0 and F = ∂F ∂w = 0 . (44)Using mass conservation, we can write the density as ρ = ˙ M / (4 πr v ). Combining the definitions of the CAK t parameter and the wind acceleration factor w , as givenin Equations (33) and (42) respectively, we can then write t as t = t m /w , where t m = v th c ˙ M Γ L ∗ (1 − Γ) . (45)This is equivalent to Equation (52) of Gayley (1995). Wenow can solve Equation (44) for w and t m to determinethe mass-loss rate.5.2. Mass-Loss Rates for the CAK Multiplier
We begin with the traditional CAK power-law form ofthe force multiplier as given in Equation 36, assumingthat the parameters α and k are known for a given set oflines. Extending beyond CAK, we also include the finitedisk factor η , which has a simple form for a CAK-likeforce multiplier when evaluated at the stellar surface: η ≈
11 + α (46)2(Kudritzki et al. 1989). Therefore we can write the crit-ical point conditions F and F as F = w + 1 − Cw α = 0 (47) F = 1 − αCw α − = 0 (48)where C = η Γ k (1 − Γ) − t − αm . Solving F for C andre-solving F for w , we find the analytic solutions w = α − α and C = 1 α α (1 − α ) − α . (49)Using this solution for C , we can solve for t m , which canthen be converted to ˙ M CAK . Combining equations (47)–(49) and recalling the definition of t m from Equation(33), we find the mass-loss rate ˙ M CAK :˙ M CAK = L ∗ (1 − Γ) v th c Γ (cid:20) α α η Γ k (1 − α ) − α (1 − Γ) (cid:21) /α . (50)It is worth mentioning here that the apparent depen-dence of ˙ M CAK on the Doppler thermal width v th is infact only a fiducial dependence, due to the definition ofthe t parameter (Equation (33)), which introduces v th and is subsequently present in Equation (45). While areformulation of t could eliminate this dependence, wechoose to carry it through our calculations in order tomaintain a level of comparability with previous works,notably that of CAK.5.3. Mass-Loss Rates for Updated Formalism
Next, we solve the critical-point equations for ourmore general form of M ( t ), given by Equation (37).Combining Equations (43) and (44), we find F and F are now given by F = w + 1 − B (cid:20) kw α ( k s w αs + ¯ Q s t αsm ) s (cid:21) = 0 (51) F = 1 − Bα (cid:20) ( kw α − )( ¯ Q s t αsm )( k s w αs + ¯ Q s t αsm ) s +1 (cid:21) = 0 (52)where B = η Γ ¯ Q/ (1 − Γ). Solving Equation (51) for¯ Q s t αsm = (cid:18) kw α Bw + 1 (cid:19) s − k s w αs , (53)we substitute the result into Equation (52). This gives0 = wB s + α ( w + 1) [( w + 1) s − B s ] . (54)In the ¯ Q → ∞ ( B (cid:29)
1) limit, Equation (54) reducesto the CAK behavior, with a solution of w = α/ (1 − α )as in Equation (49). In the opposite limit ( B → w ≈ −
1, which isunphysical (see below).For the nominal case of s = 1, Equation (54) reducesto a quadratic equation with two unique solutions for w : w = − B (1 − α )2 α − ± B (1 − α )2 α (cid:115) αB (1 − α ) (55)However, in the more general case of s (cid:54) = 1 Equation(54) must be solved numerically. In this work this isdone using the Newton-Raphson method. Once the ac-celeration factor w is known, we can then find t m fromEquation (53), which then in turn allows us to find themass-loss rate ˙ M alt from Equation (45).It is relevant to note that in calculating the mass-loss rates that result from both the CAK and alternateforms, we discard any sets of parameters that result inΓ ≥ w . In Equations (54) and (55) above, thecondition w ≥ B ≥
1. Since both η and (1 − Γ) tend to be order-unity quantities, the con-dition for physically realistic solutions is thus Γ (cid:38) ¯ Q − .With typical values of ¯ Q of a few thousand, this impliesthat whenever Γ drops below ∼ − , a steady-stateline-driven wind may not be possible.5.4. CAK Critical Point and Stellar Parameters
Although the force multiplier M ( t ) is a function ofboth T and ρ , it is possible to characterize much of thephysics by evaluating M ( t ) at the critical point of theflow (see, e.g., CAK; Abbott 1980; Pauldrach et al. 1986;Bjorkman 1995). To do this, we need to know boththe temperature and the density at the critical point.For an isothermal wind, T at the critical point is givenmore or less by the photospheric effective temperature.The only way to estimate the density ρ crit at the criticalpoint, though, is to have an associated “initial guess”for the full radial dependence of the plasma parame-ters. We provide this initial guess for a set of idealizedmain-sequence stellar properties (see Section 5.5) by us-ing a modified-CAK (mCAK) numerical code developedby Cranmer & Owocki (1995). This code solves theequations of mass and momentum conservation for thepower-law CAK force multiplier and a standard versionof the uniformly illuminated finite-disk factor η . Whenconsidering finite sound-speed effects, it is necessary tosolve simultaneous singularity and regularity conditionsfor the properties of the critical point (CAK).For a sequence of stellar properties spanning effectivetemperatures between 5,920 and 46,000 K (see below ine Driven Winds k (b)3.8 4.0 4.2 4.4 4.6 4.8log T (K)1.01.52.02.53.03.5 s (c) 3.8 4.0 4.2 4.4 4.6 4.8log T (K)10 Q (d) l o g ( g c m ) Figure 10.
Evolution with temperature of the four parameters that define the saturated power-law form of M ( t ) along theCAK critical point line. (a) α , (b) k , (c) s , (d) ¯ Q . for details), we produced mCAK models with fixed line-force constants α = 0 . k = 0 .
5. These modelsall exhibited critical points at radial distances between1.01 and 1.02 times the photospheric stellar radius, crit-ical wind speeds between about 50 and 120 km s − (i.e.,typically about 3% of the asymptotic or terminal windspeeds of 2,000–3,000 km s − ), and values of ρ crit be-tween 10 − and 10 − g cm − . Figure 9 shows thistrend in two-dimensional ( T , ρ ) diagrams. A power lawof the form ρ crit = (cid:0) . × − g cm − (cid:1) (cid:18) T K (cid:19) . (56)is reasonably successful at capturing this trend as well.Figure 10 shows how the calculated parameter ¯ Q andthe fit parameters α , k , and s vary with temperaturealong the CAK critical point curve, which correspondsto the black dashed line shown in Figure 9.For the remainder of this work we use the stellarcolor and effective temperature sequence as in Table 5 of Pecaut & Mamajek (2013) to calculate mass-loss ratesusing the methods described in above. Figure 11 showsthe continuous functions fit to the data from this tablefor both temperature-luminosity and temperature-massrelationships. These take the the form of a power-lawand a third order polynomial respectively, given bylog( L/L (cid:12) ) = 6 .
73 log( T ) − .
47 (57)andlog(
M/M (cid:12) ) = 1 .
29 log( T ) − .
44 log( T ) + 63 .
02 log( T ) − . . (58)For the purposes of this work, we assume that the windtemperature T in K remains equal to the stellar effec-tive temperature T eff . These fits were done so that mass ∼ emamajek/EEM dwarf UBVIJHK colors Teff.txt log T eff (K) log( M / M )log( L / L ) Figure 11.
Main-sequence stellar parameters from Pecaut& Mamajek (2013). Black points indicate temperature-luminosity relationship, black crosses indicate temperature-mass relationship. Functions fitted to the data points areindicated by dashed and dot-dashed lines respectively. Or-ange star indicates the location of the Sun.
Comparison of Mass-Loss Rates
Figure 12 shows a preliminary comparison of ˙ M CAK and ˙ M alt . We hold steady the parameters ¯ Q , α , k , and s , with only temperature T varying, and mass and lu-minosity dependent on temperature as described above.There is good agreement between ˙ M CAK and ˙ M alt athigh temperatures. However, there is a steep drop offexhibited at ∼ , M alt , whereas ˙ M CAK contin-ues as a power-law described by˙ M CAK ∝ T . . (59)Using Equation (57), this can also be written as˙ M CAK ∝ (cid:18) L ∗ L (cid:12) (cid:19) . . (60)This is in comparison to the common form ˙ M ∝ L /α ∗ .For a value of α = 0 . M ∝ L . ∗ , whereas Equation (60) shows aslightly weaker dependence on luminosity. In Figure log T (K) M ( M y r ) M alt M CAK
Figure 12.
Preliminary comparison of ˙ M CAK and ˙ M alt for α = 0 . k = 0 .
5, and ¯ Q = 2000. Red dashed line indicates˙ M CAK .
12, the black curve for ˙ M alt shows a strong drop-off,or quenching, which is a result of the flattening of theforce multiplier at low values of t .Next we introduce varying values of α , k , ¯ Q and s .These parameters vary according to temperature anddensity, as seen in Section 4.3. For the remainder of thiswork we consider only the version of ˙ M alt in which allfour parameters ( α , k , ¯ Q , s ) are allowed to vary withtemperature and density. Figure 13 shows the varia-tions of ˙ M alt with T and ρ . Figure 14 compares the log T (K) l o g ( g / c m ) l o g M a l t ( M y r ) Figure 13.
Contours of ˙ M alt over density and temperature.Dashed line indicates the CAK critical point density. mass-loss rates resulting from the CAK and alternateform. As in Figure 12, we see a sharp drop-off of ˙ M alt in comparison to the CAK form, commonly occurringat log T ≈ .
2. At high densities ( ρ > − g cm − )the departure from the CAK form is less pronounced, ine Driven Winds M alt between 10 , ,
000 K is a result of discarding any wind solu-tion that results in a negative wind acceleration factor w . Physically, this represents regions of the parameterspace where the wind is quenched, a phenomenon that isnot evident when M ( t ) is modeled as a pure power-lawfunction of t .Also shown in Figure 14 is the photon-tiring limit,which constrains the maximum possible mass-loss rate˙ M max (Owocki & Gayley 1997). This is limit is definedby when the kinetic energy carried away by the wind isequal to the photon energy carried by the stellar lumi-nosity, and is also dependent on the terminal velocityof the wind. If the terminal velocity v ∞ is defined as v ∞ = f v esc , then the limit ˙ M max is given by˙ M max = 2 f L ∗ v , (61)with the escape velocity given by v = 2 GM ∗ /R ∗ . Forthis work we take the traditional value of f = 3. Forlower densities ( ρ (cid:46) − g/cm − ), we see that thephoton-tiring limit intersects the mass-loss rate curvesat approximately the same point at high temperatureswhere Γ ≥ DISCUSSION AND CONCLUSIONSIn this work, we have constructed an updated list ofatomic data, with data for 4,514,900 spectral lines takenfrom the NIST, CMFGEN, CHIANTI, and TOPbasedatabases. These atomic data were then used to cal-culate the line strength parameter q i for each line for adensity range of 10 − to 10 − g cm − over a tempera-ture range of 5,200 to 70,000 K . The weighting function (cid:102) W i was also calculated. These parameters were used tofind the line force multiplier M ( t ) over a range of t from10 − to 10. The distribution of M ( t ) was fit usinga power-law as described by Equation (36), as well asan alternate function in the form of a saturated power-law, as described by Equation (37). We found that thisalternative function better describes the values of theline-force multiplier as calculated from the updated linelist, especially at low values of t . The residuals of thisalternate function are consistently lower than those thatresult from the CAK form in the case of low t , and com-parable for high values of t . This is consistent acrosstemperatures and densities. We include the parame-ter s to control the sharpness of the turn-over from thepower-law segment to the flat segment. M alt ( t ) reduces Atomic data and other parameters calculated in the course of thiswork are available at https://github.com/aslyv2/Rad-Winds to the power-law form in the limit of high- t for all val-ues of the sharpness parameter s. In the limit of low- tM alt ( t ) similarly reduces to the calculated value of ¯ Q .Using the alternate function for M ( t ), we also cal-culate mass-loss rates for the temperatures and densi-ties in our grid, using the fitted parameters α, k, and s ,along with the corresponding calculated values of ¯ Q . Wefind that the sharpness parameter s has a drastic effecton the determined mass-loss rates, especially at hightemperatures. Additionally, there is a sharp drop-off inthe mass-loss rates calculated from the updated formof M ( t ) and a resulting discrepancy between it and theCAK mass-loss form. This drop-off in the mass-loss ratedescribes a quenching of the line-driven wind that is notpresent in the CAK form.We find that the quenching of the wind typically oc-curs between temperatures of 10 ,
000 K and 20 ,
000 Kand at luminosities of 2 . (cid:46) log( L ∗ /L (cid:12) ) (cid:46) .
75. Thismay be a partial explanation for the discrepancy notedbetween empirically derived mass-loss rates and pre-dicted values for stars of luminosities below ∼ L (cid:12) ( T ≈ ,
000 K), often referred to in the literature as the“weak-wind problem” (Muijres et al. 2012), although itshould be noted that our calculations place the quench-ing of the wind at lower luminosities and temperatures.It is also possible that these effects could be importantto include when modeling the oscillations of Slowly Pul-sating B (SPB) stars, which have T eff values betweenabout 10,000 and 20,000 K (De Cat 2007). The interac-tions between their oscillations and winds remain poorlyunderstood (e.g., Saio 2015).Lastly, there is another physical effect that must betaken into account to fully understand how the pre-dicted quenching effect manifests itself: collisionless de-coupling between the line-driven ions and the dominanthydrogen/helium gas. This has been proposed to beimportant both for B-type stars (Springmann & Paul-drach 1992; Babel 1996; Krtiˇcka 2014) and some metal-enriched AGNs (Baskin & Laor 2012). In some low-density systems this decoupling can lead to frictionalheating with wind temperatures far in excess of thestellar T eff , and in others may produce fully-separatedmulti-component winds with peculiar chemical abun-dances. It may be possible that the drastic reductionin the ion line-force (which arises due to the flatten-ing of the force multiplier) allows these systems to un-dergo a more gentle and gradual transition from a cou-pled single-fluid outflow to a quiescent hydrostatic at-mosphere.Although here we consider only the assumption ofLTE, other similar works consider the effects of NLTE(see, for example, Gormaz-Matamala et al. 2019). Puls6 M ( M y r ) M alt M CAK M max M ( M y r ) M ( M y r ) log T (K) M ( M y r ) log T (K) log T (K) l o g ( g c m ) Figure 14.
Comparison of ˙ M CAK and ˙ M alt , by density order of magnitude. Dashed lines indicate ˙ M CAK , solid lines indicate˙ M alt . Lower right corner shows calculations of ˙ M CAK and ˙ M alt for the calculated critical point densities. Red dashed linesindicate the photon-tiring limit. et al. (2000) accounted for NLTE effects in the line dis-tribution by restricting the types of lines used to thosewith or directly connected to those with a ground ormetastable lower level. For our purposes, it will be use-ful to refine the ionization balance used here using themodified nebular approach described by others (see, forexample, Abbott & Lucy 1985, Gormaz-Matamala et al.2019). Although the assumption of a Planck function for F ( ν ) allowed us to maintain generality in this work,in future work it will be necessary to refine our choiceof F ( ν ) to a more realistic distribution. For example,a self-consistent treatment of absorption in the near-star atmosphere could be applied to the phenomenon of“bistability” (e.g Lamers & Pauldrach 1991) in whichthe wind sees a lower flux shortward of 91.2 nm—anda higher flux in the Balmer continuum—and this af- ine Driven Winds M ( t ). Alternatively, this bistability jump could bea result of the recombination of Fe between Fe III andFe IV, with the contribution of the Fe III lines dominat-ing the radiative acceleration of the subsonic part of thewind (Vink et al. 1999; Vink 2000). Puls et al. (2000)similarly found that at low line strengths mass-loss isdominated largely by the radiative acceleration of irongroup elements, with lighter ions playing a more impor-tant role at larger line strength. This bistability jump ispredicted to be reflected by an increase in mass loss, oc-curring around ∼ k parameter (seeFigure 9(a)). However, while the mass-loss rate is usu-ally quite sensitive to k , we do not see any significantincrease in our final calculations for ˙ M at these param-eters (e.g., Figure 14).In this study we have also limited ourselves to the solarelemental abundances of Asplund et al. (2009). Otherabundance patterns, such as those found in nearbygalaxies with lower metallicity (Puls et al. 1996) or incertain types of chemically peculiar stars (Alecian & Stift 2019) should be explored. Additionally, we planto explore the radial dependence of the t parameter andthe associated spatial variation of M ( t ) in self-consistentmodels of radial outflow from stars and other luminousastrophysical sources such as active galactic nuclei.ACKNOWLEDGMENTSCHIANTI is a collaborative project involving GeorgeMason University, the University of Michigan (USA),University of Cambridge (UK) and NASA GoddardSpace Flight Center (USA). This work was supportedby start-up funds from the Department of Astrophys-ical and Planetary Sciences at the University of Col-orado Boulder. This research made use of NASA’s As-trophysics Data System (ADS). The authors would alsolike to thank the anonymous referee for their helpfulcomments. Software:
Python v3.7.6 (Van Rossum & Drake2009), NumPy (Oliphant 2006; Van Der Walt et al.2011), SciPy (Virtanen et al. 2020), matplotlib (Hunter2007), AstroPy (Astropy Collaboration et al. 2013; Price-Whelan et al. 2018)APPENDIX A. DATABASE SELECTION AND SPECIFIC LINE COUNTSIn cases where more than one database listed atomic data for a given ion, the database with the largest number ofavailable transitions was used for each ionization state of each element. In general, CMFGEN was found to contain themost lines for a majority of ions. However, where CMFGEN data was nonexistent or insufficient, the database withthe next most lines was used. Generally, this was CHIANTI. For several elements and ionization states (namely NVI, N VII, Cl VIII, Cl IX, and Ni X), the necessary atomic data was not available from the databases used. Table A1shows a breakdown of line counts n by ion, with the database used for each also listed. In some databases, transitionswith a both a radiative decay rate and an autoionization rate were represented twice. After compiling the line listfrom the total available data, we discarded any such duplicate transitions. Table A1 . Number of lines ( n ) and database used for each ion. A dash (-) indicates that no data were available. CMFGEN,NIST, CHIANTI, and TOPbase are abbreviated as CM, N, CH, and T respectively. Ion n Database Ion n Database Ion n Database Ion n DatabaseH I 435 CM Na V 10644 CM Cl IV 8612 CM V III 21 NHe I 3857 CM Na VI 10994 CM Cl V 3388 CM V IV 239 NHe II 435 CM Na VII 5436 CM Cl VI 2377 CM V V 10 NLi I 68 N Na VIII 4742 CM Cl VII 1557 CM V VI 4 NLi II 134 N Na IX 4201 CH Cl VIII - - V VII 7 NLi III 2 N Na X 331 CH Cl IX - - V VIII 6 NBe I 175 N Mg I 2841 CM Cl X 24 CH V IX 16 N
Table A1 continued Table A1 (continued)
Ion n Database Ion n Database Ion n Database Ion n DatabaseBe II 97 N Mg II 2641 CH Ar I 3824 CM V X 13 NBe III 100 N Mg III 4753 CH Ar II 79388 CM Cr I 49885 CMBe IV 10 N Mg IV 5706 CM Ar III 6901 CM Cr II 66400 CMB I 96 N Mg V 6377 CM Ar IV 11290 CM Cr III - -B II 150 N Mg VI 14480 CM Ar V 8350 CM Cr IV 67061 CMB III 74 N Mg VII 11940 CM Ar VI 5 N Cr V 43860 CMB IV 134 N Mg VIII 5820 CM Ar VII 35 CH Cr VI 4406 CMB V 58 N Mg IX 5517 CM Ar VIII 2743 CH Cr VII 46 CHC I 10204 CM Mg X 26078 CH Ar IX 5691 CH Cr VIII 131 CHC II 8017 CM Al I 4985 CM Ar X 4435 CH Cr IX 236 CHC III 9468 CM Al II 2870 CM K I 1471 CM Cr X 16 NC IV 1297 CM Al III 2665 CM K II 38603 CM Mn I 164 NC V 2196 CM Al IV 5296 CH K III 220 CM Mn II 49066 CMC VI 1575 CM Al V 6607 CH K IV 18227 CM Mn III 70218 CMN I 855 CM Al VI 7989 CM K V 7252 CM Mn IV 72374 CMN II 7879 CM Al VII 15486 CM K VI 14870 CM Mn V 77009 CMN III 6710 CM Al VIII 13501 CM K VII 71 N Mn VI 70116 CMN IV 13886 CM Al IX 5859 CM K VIII 95 N Mn VII 8277 CMN V 1296 CM Al X 5041 CM K IX 2758 CH Mn VIII 47 CHN VI 2263 CM Si I 2791 CM K X 5700 CH Mn IX 137 CHN VII 3150 CM Si II 4196 CM Ca I 106 N Mn X 236 CHO I 4145 CM Si III 1328 CM Ca II 238 CH Fe I 141928 CMO II 17874 CM Si IV 2672 CH Ca III 520 N Fe II 530827 CMO III 6516 CM Si V 5354 CH Ca IV 2 N Fe III 136060 CMO IV 7599 CM Si VI 6518 CM Ca V 9 CH Fe IV 72223 CMO V 3237 CM Si VII 9364 CM Ca VI 10 CH Fe V 71983 CMO VI 1569 CM Si VIII 705 CH Ca VII 86 CH Fe VI 185392 CMO VII 3505 CM Si IX 403 CH Ca VIII 200 CH Fe VII 86504 CMO VIII 1575 CM Si X 5017 CH Ca IX 9230 CH Fe VIII 21134 CHF I 119 N P I 46 N Ca X 2760 CH Fe IX 47085 CHF II 2354 CM P II 217043 CM Sc I 259 N Fe X 50854 CMF III 9725 CM P III 5576 CM Sc II 77253 CM Co I 118 NF IV 15 N P IV 2537 CM Sc III 687 CM Co II 61986 CMF V 11 N P V 2700 CH Sc IV 4 N Co III 679412 CMF VI 8415 T P VI 5533 CH Sc V 4 N Co IV 69425 CMF VII 6406 T P VII 3 CH Sc VI - - Co V 75923 CMF VIII 5614 T P VIII 25 CH Sc VII 15 N Co VI 75118 CMF IX 3488 T P IX 59 CH Sc VIII 16 N Co VII 68388 CMNe I 2629 CM P X 78 CH Sc IX 15 N Co VIII 88548 CMNe II 5795 CM S I 19813 CM Sc X - - Co IX 12232 CMNe III 2343 CM S II 8527 CM Ti I 490 N Co X 5 NNe IV 9725 CM S III 4543 CM Ti II 93118 CM Ni I 188 NNe V 13037 CM S IV 7530 CM Ti III 21722 CM Ni II 51812 CMNe VI 5171 CM S V 3605 CM Ti IV 1000 CM Ni III 66511 CMNe VII 5213 CM S VI 1936 CM Ti V 4 N Ni IV 72898 CMNe VIII 26832 CH S VII 73 N Ti VI 11 N Ni V 75541 CMNe IX 216 CH S VIII 54 N Ti VII 1 N Ni VI 79169 CM
Table A1 continued ine Driven Winds Table A1 (continued)
Ion n Database Ion n Database Ion n Database Ion n DatabaseNe X 190 CH S IX 51 N Ti VIII 15 N Ni VII 74411 CMNa I 2778 CM S X 57 N Ti IX 14 N Ni VIII 71614 CMNa II 5054 CH Cl I 75 N Ti X 43 N Ni IX 79227 CMNa III 4368 CH Cl II 52 N V I 1095 N Ni X - CMNa IV 3754 CM Cl III 50 N V II 1415 N - - -
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