Simulations to test He-like forbidden-to-intercombination line ratio modeling for O, Ne, Mg, S, Si and Fe. June 28, 2005 M. Walter-Range Swarthmore College

Here are some of the plots of R vs. n e from mid-May, but I have updated them to include the analytic expression R=R 0 /(1+n e /n c ) using Porquet’s newer values for R 0. According to Equation 4 in Blumenthal, Drake, and Tucker 1972,. The values of n c are from BDT: Oxygen: n c =3.4E10, Neon: n c =6.4E11, Magnesium: n c =6.2E12, Silicon: n c =4.0E13. The PrismSPECT modeling uses ATBASE v4.3 beta, with the “most detailed” model chosen for the He-like atomic models. The workspaces and the associated ATM files can be found at:

Porquet’s group did not recalculate R 0 for any elements heavier than Si, so the following two plots only contain the original BDT line and the PrismSPECT values. For Sulfur: n c =1.9E14, Iron: n c =4.7E16. The largest disagreement between PrismSPECT and the BDT function occurs in the low-density limit for each of the elements shown above. However, the calculations do show agreement with the analytic expression in terms of overall shape. This includes the (effective) values for n c in PrismSPECT/ATBASE. There is significantly less agreement for plots of R vs. φ, shown on the following pages. Questions: (i) What is the cause of the discrepancy between R 0 in PrismSPECT and the values calculated by BDT and Porquet? (ii) Can we be sure that the atomic rates governing R 0 are more accurate in the PrismSPECT/ATBASE models? (iii) How accurate is the iron model (both in terms of n c and R 0 )?

The following plots are the results from simulations in which the varying property was the radiation source drive temperature (and thus the UV mean intensity driving the 3 S – 3 P photoexcitation). I used a one-sided radiation field, and the planar option for the plasma geometry (I also tried the zero-width option, but it didn’t seem to make any difference). If you would like to see the exact settings I entered into PrismSPECT, there are PowerPoint documents with screenshots for all the simulations I ran, available at: I have also uploaded several of the workspaces I was using. They can be found at: The step-by-step procedure I used is available at: This procedure is based on equations that can be found at: Please look this over to make sure you agree with the mathematical formalism relating φ, R * /r, and T rad. For each ion, there are two plots. The first is a simple comparison of PrismSPECT results and the BDT model (using Porquet’s newer values for R 0, although I did include the original BDT model as a dotted line). The red line segment and data points are for values of φ we would expect to find when the radial distance of the plasma (from the center of the star) is between 1R * and 100R *. The second plot for each element shows explicitly the relationship between the f/i ratio and the radial distance.

This is a plot of the Tlusty model atmosphere at the at the relevant range of wavelengths for Si XIII. The vertical line shows the wavelength I used as the weighted average for the two 2 3 S P transitions. Note: ATBASE says these two transitions are at Å and Å. Then I found the mean flux over a 5-Angstrom range in order to account for the Doppler shift of the stellar wind. The effective value of H is thus 4.69E-4 ergs s -1 cm -2 Hz -1. This particular Tlusty model is for a 30kK star with log(g)=3.25 and Z/Z 0 =1. I have also plotted a 30kK blackbody curve for comparison, and blackbodies for the temperatures shown in Table 2 of the document, “phi_equations.pdf.” This was done in order to check and see if the radiation temperatures I was giving PrismSPECT were scaling sensibly according to the dilution factor, W(r). It seems that they are. Check Table 2 in the document for the associated values of R * /r, W, J, and φ.

T plasma =500eV which results in 89% ionization to Si XIII. T spec =29.2kK=2.520eV Note: This is the calculated radiation temperature at R * /r=1, i.e. the surface of the star. A blackbody curve for this temperature should intersect a 30kK Tlusty model atmosphere at Å This is the (average) λ uv that drives the 2 3 S P transitions. H ν = 4.69E-4 ergs s -1 cm -2 Hz -1. φT rad (eV)f/i E E E E E E E E E-30 1E E-30 The solid line in the plot is the BDT model, calculated with Porquet’s newer values for R 0. The original BDT line (using values of R 0 from 1972) is shown as a dotted line. The individual points are the results of PrismSPECT simulations. A black point meant that I was calculating R at a specific value of φ, a red point meant that I was calculating R for a specific value of u (u=R * /r). The following page displays a plot of R vs. u, using these red points and the red segment of the BDT line (which corresponds to radial distances of 1R * <r<100R * ). My primary reason for highlighting this segment was so that I could see which values of φ were more likely to be observed in an astronomical context.

The solid line is the BDT model, calculated with Porquet’s newer values of R 0. The points are from PrismSPECT simulations. Note: see “phi_equations.pdf” for a description of how we relate u to φ and to T rad. In this case, I have also plotted the BDT line from a pre-print copy of Maurice Leutenegger’s paper on He-like triplet ratios. We used the same model atmosphere, but we used slightly different methods for finding the mean intensity of the radiation, hence the difference between my line and his. A copy of his paper can be found at: It is also interesting to note that the BDT line and the PrismSPECT line appear to have significantly different shapes, even though they are in agreement at the end points. This disagreement can also be seen in the R vs. φ plot on the previous page.

T plasma =500eV which results in 55% ionization to Mg XI. T spec =32.8kK=2.828eV λ uv(ave) = Å H ν = 1.14E-3 ergs s -1 cm -2 Hz -1. φT rad (eV)f/i E E E E E E E E E E-30 1E E-30 The solid line in the plot is the BDT model, calculated with Porquet’s newer values for R 0. The original BDT line (using values of R 0 from 1972) is shown as a dotted line. The individual points are the results of PrismSPECT simulations. A black point meant that I was calculating R at a specific value of φ, a red point meant that I was calculating R for a specific value of u (u=R * /r). The following page displays a plot of R vs. u, using these red points and the red segment of the BDT line (which corresponds to radial distances of 1R * <r<100R * ).

The solid line is the BDT model, calculated with Porquet’s newer values of R 0. The points are from PrismSPECT simulations.

T plasma =500eV which results in 89% ionization to S XV. T spec =29.2kK=2.520eV λ uv(ave) = Å H ν = 2.31E-3 ergs s -1 cm -2 Hz -1. φT rad (eV)f/i E E E E E E E E E E E-30 1E E-30 The solid line in the plot is the BDT model, calculated with the original (1972) value for R 0 because Porquet’s group did not publish an updated value for S XV. The individual points are the results of PrismSPECT simulations. A black point meant that I was calculating R at a specific value of φ, a red point meant that I was calculating R for a specific value of u (u=R * /r). The following page displays a plot of R vs. u, using these red points and the red segment of the BDT line (which corresponds to radial distances of 1R * <r<100R * ).

The solid line is the BDT model, using the 1972 value of R 0. The points are from PrismSPECT simulations.

T plasma =100eV which results in 95% ionization to O VII. T spec =27.6kK=2.374eV λ uv(ave) = Å H ν = 9.69E-4 ergs s -1 cm -2 Hz -1. φT rad (eV)f/i E E E E E-30 1E E E E E E E E-30 The solid line in the plot is the BDT model, calculated with Porquet’s newer values for R 0. The original BDT line (using values of R 0 from 1972) is shown as a dotted line. The individual points are the results of PrismSPECT simulations. A black point meant that I was calculating R at a specific value of φ, a red point meant that I was calculating R for a specific value of u (u=R * /r). The following page displays a plot of R vs. u, using these red points and the red segment of the BDT line (which corresponds to radial distances of 1R * <r<100R * ). I initially ran this simulation with an ion density of cm -3, which is close to the value of n c for oxygen. Realizing my mistake, I repeated the simulation with a density of 10 7 cm -3, but this change barely made a difference. For instance, the f/i ratio at φ=1 changed from 0.51 to 0.53, which is negligible when compared with the disparity between BDT’s value and the one from PrismSPECT. I tried a simulation with φ=1E-125 (T rad =0.025eV) and I recovered the same value as the one shown in the low-density limit of the R vs. n e plot.

The solid line is the BDT model, calculated with Porquet’s newer values of R 0. The points are from PrismSPECT simulations.

Conclusions: In plots of R vs. n e, the largest disparity between BDT and PrismSPECT occurs at low densities, most noticeably in the case of Fe XXV. What rates could account for these differences? We have developed a formalism and procedure for relating BDT’s analytic expression for R to the results of PrismSPECT models, relating φ to u=R * /r and T rad. Do you agree with this formalism? There is significant disagreement between PrismSPECT models of the effects of photoexcitation and the models of BDT. Could you check the actual photoexcitation rates in PrismSPECT in one of these calculations? Are other procedures (not accounted for by BDT) important? Are crucial rates significantly different?