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Model Photospheres I. What is a photosphere? II. Hydrostatic Equilibruium III.Temperature Distribution in the Photosphere IV. The Pg-Pe-T relationship V. Properties of the Models VI. Models for cool stars I. What is a Stellar Atmosphere? • Transition between the „inside“ and „outside“ of the star • Boundary between the stellar interior and the interstellar medium • All energy generated in the core has to pass through the atmosphere • Atmosphere does not produce any energy • Two basic parameters: • Effective temperature. Not a real temperature but the temperature needed to produce the observed flux via 4pR2T4 • Surface Gravity – log g (although g is not a dimensionless number). Log g in stars range from 8 for a white dwarf to 0.1 for a supergiant. The sun has log g = 4.44 What is a Photosphere? • It is the surface you „see“ when you look at a star • It is where most of the spectral lines are formed What is a Model Photosphere? • It is a table of numbers giving the source function and the pressure as a function of optical depth. One might also list the density, electron pressure, magnetic field, velocity field etc. • The model photosphere or stellar atmosphere is what is used by spectral synthesis codes to generate a synthetic spectrum of a star Real Stars: 1. Spherical 2. Can pulsate 3. Granulation, starspots, velocity fields 4. Magnetic fields 5. Winds and mass loss Our model: 1. Plane parallel geometry 2. Hydrostatic equilibrium and no mass loss 3. Granulation, spots, and velocity fields are represented by mean values 4. No magnetic fields II. Hydrostatic equilibrium dA P +dP P + dP P r + dr A dr r dm P M(r) Gravity The gravity in a thin shell should be balanced by the outward gas pressure in the cell Fp = PdA –(P + dP)dA = –dP dA dM FG = –GM(r) r2 Pressure Force Gravitational Force P +dP r M(r) = ∫ r(r) 4pr2 dr A 0 dr dM = r dA dr Both forces must balance: dm FP + FG = 0 –dP dA + –G r(r)M(r) dA dr r2 P =0 Gravity The pressure in this equation is the total pressure supporting the small volume element. In most stars the gas pressure accounts for most of this. There are cases where other sources of pressure can be significant when compared to Pg. Other sources of pressure: 1. Radiation pressure: PR = 2. Magnetic pressure: 4s T4 3c = 2.52 ×10–15 T4 dyne/cm2 B2 PM = 8p 3. Turbulent pressure: ~ ½rv2 v is the root mean square velocity of turbulent elements Footnote: Magnetic pressure is what is behind the emergence of magnetic „flux ropes“ in the Sun Pphot B2 Ptube+ 8p In the magnetic flux tube the magnetic field provides partial pressure support. Since the total pressure in the flux tube is the same as in the surrounding gas Ptube < Pphot. Thus rtube < rphot and the flux tube rises due to buoyancy force. Pressures Teff (K) Sp Pg (dynes/cm2) PR (dynes/cm2) B (Gauss) v 4000 K5 V 1 × 105 0.6 1584 7.5 8000 A6 V 1 × 104 10 501 10.6 12000 B8 V 3 × 103 52 274 13.0 16000 B3 V 3 × 103 165 274 15.0 20000 B0 V 5 × 103 403 354 16.7 B is for magnetic pressure = Pg v is velocity that generates pressure equal to Pg according to ½rv2 We are ignoring magnetic fields in generating the photospheric models. But recall that peculiar A-type stars can have huge global magnetic fields of several kilogauss in strength. In these atmospheric models one has to treat the magnetic pressure as well. dP = –G r(r)M(r) dr r2 In our atmosphere GM/r2 = g (acceleration of gravity) F x increases inward so no negative sign P dx P + dP dA r F + dF dP = grdx The weight in the narrow column is just the density × volume × gravity dtn = knrdx gravity dP = g dtn kn One way to integrate the hydrostatic equation Pg½ dPg = Pg½ g dk0 k0 k0 is a reference wavelength (5000 Å) to Pg(t0) = ( 3 g 2 0∫ log to Pg(t0) = ( ⅔ Pg½ dt0 k0 t0½ Pg½ 3 g ∫ k log e dlog t0 2 –∞ 0 ⅔ ( 2 P (t ) = 3 g 0 0∫ gPg½ dt0 k0 ( to 3/2 Integrating on a logarithmic optical depth scale gives better precision Numerical Procedure • Guess the function Pg(t0) and perform the numerical integration • New value of Pg(t0) is used in the next iteration until convergence is obtained • A good guess takes 2-3 iterations Problem: we must now k0 as a function of t0 since k0 appears in the integrand. kn is dependent on temperature and electron pressure. Thus we need to know how T and Pe depend on t0. III. Temperature Distribution in the Solar Photosphere Two probes of depth: q • Limb Darkening ds The increment of path length along the line of sight is ds = dx sec q • Wavelength dependence of the absorption coefficient Limb darkening is due to the decrease of the continuum source function outwards ∞ In (0) = 0 ∫Sne–tnsec q sec q dtn The exponential extinction varies as tnsec q, so the position of the unit optical depth along the line of sight moves upwards, i.e. to smaller t. Limb Darkening Temperature profile of photosphere Temperature Bottom of photosphere 10000 8000 6000 4000 q2 q1 z=0 tn =1 surface Top of photosphere z dz The path length dz is approximately the same at all viewing angles, but at larger the optical depth of t=1 is reached higher in the atmosphere Solar limb darkening as a function of wavelength in Angstroms Solar limb darkening as a function of position on disk At 1.3 mm the solar atmosphere exhibits limb brightening Horne et al. 1981 In radio waves one is looking so high up in the atmosphere that one is in the chromosphere where the temperature is increasing with heigth Temperature Temperature profile of photosphere and chromosphere 10000 8000 chromosophere 6000 4000 z=0 z Limb darkening in other stars Use transiting planets No limb darkening transit shape At the limb the star has less flux than is expected, thus the planet blocks less light The depth of the light curve gives you the Rplanet/Rstar, but the „radius“ of the star depends on the limb darkening, which depends on the wavelength you are looking at q To get an accurate measurement of the planet radius you need to model the limb darkening appropriately If you define the radius at which the intensity is 0.9 the full intensity: At l=10000 Å, cos q=0.6, q=67o, projected disk radius = sin q = 0.91 At l=4000 Å, cos q=0.85, q=32o, projected disk radius = sin q = 0.52 → disk is only 57% of the „apparent“ size at the longer wavelength The transit duration depends on the radius of the star but the „radius“ depends on the limb darkening. The duration also depends on the orbital inclination When using different data sets to look for changes in the transit duration due to changes in the orbital inclination one has to be very careful how you treat the limb darkening. Possible inclination changes in TrEs-2? Evidence that transit duration has decreased by 3.2 minutes. This might be caused by inclination changes induced by a third body But the Kepler Spacecraft does not show this effect. One possible explanation is that this study had to combine different data sets taken at different wavelength band passes (filters). But the limb darkening depends on wavelength. At shorter wavelengths the star „looks“ smaller. The only star for which the limb darkening is well known is the Sun In the grey case had a linear source function: Sn = a + btn Using: ∞ In (0) = 0 ∫Sne–tnsec q sec q dtn In (0) = a + b cosq This is the Eddington-Barbier relation which says that at cos q = tn the specific intensity on the surface at position equals the source function at a depth tn Limb darkening laws usually of the form: Ic = Ic(0) (1 – e + e cos q) e ≈ 0.6 for the solar case, 0 for A-type stars Ic(0) continuum intensity at disk center Rewriting on a log scale: ∞ In (0) = –tnsec q ∫S e n 0 d log tn tn sec q log e Contribution function No light comes from the highest and lowest layers, and on average the surface intensity originates higher in the atmosphere for positions close to the limb. Sample solar contribution functions Wavelength Variation of the Absorption Coefficient Since the absorption coefficient depends on the wavelength you look into different depths of the atmosphere. For the Sun: • See into the deepest layers at 1.6mm • Towards shorter wavelengths kn increases until at l = 2000 Å it reaches a maximum. This corresponds to a depth of formation at the temperature minimum (before the increase in the chromosphere) Solar Temperature distributions Best agreements are deeper in the atmosphere where log t0 = –1 to 0.5 Poor aggreement is higher up in the atmosphere Temperature Distribution in other Stars The simplest method of obtaining the temperature distribution in other stars is to scale to a standard temperature distribution, for example the solar one. T(t0) = S0T(סּt0) In the grey case: ¼ T(t) = ¾ [t + q(t)] Teff Teff T(t) = Tסּ T(סּt) eff In the grey case the scaling factor is the ratio of effective temperatures Scaled solar models agree well (within a few percent) to calculations using radiative equilibrium. They also agree well when applying to giant stars. Numerically it was easier to use scaled solar models in the past. Now, one just uses a grid of models calculated using radiative equilibrium IV. The Pg–Pe–T Relation When solving the hydrostatic pressure equation we start with an initial guess for Pg(t0). We then require that the electron pressure Pe(t0) = Pe(Pg) in order to find k0(t0) = k0(T,Pe) for the integrand. The electron pressure depends on the temperature and chemical composition. N1j N0j = Fj(T) Pe N1j = number of ions per unit volume of the jth element See Saha equation from 2nd lecture N0j = number of neutrals F(T) = 0.65 u1 u0 T 5/2 10–5040I/kT Neglect double ionization. N1j = Nej, the number of electrons per unit volume that are contributed by the jth element. Fj(T) Pe = Nej N0j = Nej Nj – Nej The total number of jth element particles is Nj = N1j + N0j. Solving for Nej Fj(T)/Pe Nej = N j 1 + F (T)/P j e The pressures are: Pe = Sj NejkT Pg = S (Nej + Nj)kT j Taking ratios: Pe Pg Pe Pg = S Nj = S NejkT S (Nej + Nj)kT Fj(T)/Pe 1 + Fj(T)/Pe Fj(T)/Pe Using the number abundance Aj = Nj/NH NH = number of hydrogen S Nj 1 + 1 + F (T)/P j e S Aj Pe = Pg Fj(T)/Pe 1 + Fj(T)/Pe Fj(T)/Pe S Aj 1 + 1 + F (T)/P j e This is a transcendental equation in Pe that has to be solved iteratively. F(T) are constants for such an iteration. Pe and Pg are functions of t0. This equation is solved at each depth using the first guess of Pg(t0). log t 1 0 –1 –2 –3 –4 For the cooler models the temperature sensitivity of the electron pressure is very large with d log Pe/d log T ≈ 12 since the absorption coefficient is largely due to the negative hydrogen ion The absorption coefficient kn is largely due to the negative hydrogen ion which is proportional to Pe so the opacity increases very rapidly with depth. Hydrogen dominates at high temperatures and when it is fully ionized Pg ≈ 2Pe At cooler temperatures Pe ~ Pg½ Where does the later come from? Assume the photosphere is made of single element this simplifies things: Pe = P g F(T)/Pe 1 + 2 F(T)/Pe Pe = F(T)Pg – 2F(T)Pe = F(T)(Pg– 2Pe) 2 Pg >> Pe in cool stars 2 Pe ≈ F(T)Pg Completing the model Pg(t0) = ( t0½ Pg½ 3 g dlog t0 ∫ 2 –∞ k0 log e ⅔ ( log to We can now can compute this • Take T(t0) and our guess for Pg(t0) • Compute Pe(t0) and k0(t0) • Above equation gives new Pg(t0) • Iterate until you get convergence (≈ 1%) • Can now calculate geometrical depth and surface flux The Geometric Depth We are often interested in the geometric depth scale (i.e. where the continuum is formed). This can be computed from dx = dt0/k0r t0 x(t0) = ∫ 1 k0(t0)r(t0) dt0 0 The density can be calculated from the pressure ( P = (r/m)KT ) r = NH (hydrogen particles per cm3) x SAjmj grams/H particle) where mj is the atomic weight of the jth element NH = N – Ne S Aj = Pg – Pe kT S Aj log t0 x(t0) = ∫ –∞ S AjkT(t0)t0 d log t0 k0(t0)S Ajmj[Pg(t0)t0 – Pe(t0)] d log e A more interesting form is to integrate on a Pg scale with dPg = rgdx 1 x(t0) = g Pg ∫ –∞ S AjkT(p) dp p S Ajmj The thickness of the atmosphere is inversely proportional to the surface gravity since T(Pg) depends weakly on gravity This makes physical sense if you recall the scale height of the atmosphere: Scale height H = kT/mg Computation of the Spectrum The spectrum ∞ Fn(0) = 2p ∫ S (t ) E (t )dt n n 2 n n 0 It is customary to integrate on a log t scale ∞ Fn(0) = 2p ∫ S (t ) E (t ) n n –∞ Flux contribution function 2 n kn(t0)t0 d log t0 k0(t0) d log e Flux Contribution Functions as a Function of Wavelength Flux at 8000 Å originates higher up in the atmosphere than flux at 5000 or 3646 Å But cross the Balmer jump and the flux dramatically increases. This is because there is a sharp decrease in the opacity across the Balmer jump. Flux Contribution Functions as a Function of Effective Temperature T= 10400 K T= 8090 K T= 4620 K A hotter star produces more flux, but this originates higher up in the atmosphere Computation of the Spectrum There are other techniques for computing the flux → Different integrals. Integrating flux equation by parts: ∞ ∫ Fn(0) = 2p Sn(tn) E2(tn)dtn 0 ∞ Fn(0) = pSn(0) + ∫ 0 d Sn(tn) E (t )dt 3 n n dtn The flux arises from the gradient of the source function. Depths where dS/dt is larger contribute more to the flux V. Properties of Models: Pressure d log T/d log Pg = 0.4 Temperature Convection gradient Relationship between pressure and temperature for models of effective temperatures 3500 to 50000 K. The dashed line marks where the slope exceeds 1–1/g ≈ 0.4 and implies instability to convection Cannot scale T(Pg), unlike T(t0)!!! Teff = 8750 K Effects of gravity dlogPg dlog g dlogPg dlog g 0.62 = = 0.85 Increasing the gravity increases all pressures. For a given T the pressure increases with gravity Pg(t0) = ( t0½ Pg½ 3 g dlog t0 ∫ 2 –∞ k0 log e ⅔ ( log to Pg ≈ C(T) g ⅔ since pressure dependence in the integral is weak. So dlog P/dlog g ~ 0.67 In general Pg ~ gp In cool models p ranges from 0.64 to 0.54 in going from deep to shallow layers In hotter models p ranges from 0.85 to 0.53 in going from deep to shallow layers Recall Pe ≈ Pg½ in cool stars → Pe ≈ constant g⅓ Pe ≈ 0.5 Pg in hot stars → Pe ≈ constant g⅔ Properties of Models: Chemical Composition Gas pressure Electron pressure • In hot models hydrogen takes over as electron donor and the pressures are indepedent of chemical composition • In cool models increasing metals → increasing number of electrons → larger continous absorption → shorter geometrical penetration in the line of sight → gas pressure at a given depth decreases with increasing metal content Qualitatively: Using SAj for the sum of the metal abundances Pe 1 Pg S Aj = = Pe NH Pg S Nj Pe PH Pg S N jkT kT kT ≈ Pe S N jkT Since PH, the partial pressure of Hydrogen dominates the gas pressure SNj = S(N1 + N0)j, the number of element particles is the sum of ions and neutrals and Pe=NekT = SN1jkT for single ionizations Pe 1 Pg S Aj ≈ S N 1j S (N 1 + N0)j In the solar case metals are ionized SN1j >> N0j dPg = Pe 1 Pg S Aj g dk0 = k0 ≈ g Pek0/Pe dk0 1 g = P SA k /P g j 0 e k0 is dominated by the negative hydrogen ion, so k0/Pe is independent of Pe Integrating: 2 1 ½Pg = S Aj t0 g ∫ k /P 0 dt0 e 0 g and T are constants –½ Pg =c0 (S Aj) ½ Pe =c0 (S Aj) For metals being neutral: SN1J << S(N1 + N0)j can show –⅓ Pg =c0 (S Aj) ⅓ Pe =c0 (S Aj) Properties of Models: Effective Temperature Note scale change of ordinate • In hotter models opacity increases dramatically • More opacity → less geometrical penetration to reach the same optical depth • We see less deep into the stars → pressure is less • But electron pressure increases because of more ionization This is seen in the models If you can see down to an optical depth of t ≈ 1, the higher the effective temperature the smaller the pressure Properties of Models: Effective Temperature For cool stars on can write: Pe ≈ C eWT At high temperatures the hydrogen (ionized) has taken over as the electron donor and the curves level off A grid of solar models Temperature Log t Depth (km) Depth (km) The mapping between optical depth and a real depth Log t Amplitude (mmag) Why do you need to know the geometric depth? Wavelength (Ang) In the case the pulsating roAp stars, you want to know where the high amplitude originates Pgas Log (Pressure) Pe ≈Pg½ Pelectron Log t Note bend in Main Sequence at the low temperature end. This is where the star becomes fully convective VI. Models for Cool Objects Models for very cool objects (M dwarfs and brown dwarfs) are more complicated for a variety of reasons, all related to the low effective temperature: 1. Opacities at low temperatures (molecules, incomplete line lists etc.) not well known 2. Convection much stronger (fully convective) 3. Condensation starts to occur (energy of condensation, opacity changes) 4. Formation of dust 5. Chemical reactions (in hotter stars the only „reactions“ are ionization which is give by ionization equilibrium) Much progress in getting more complete line lists for water as well as molecules. Models have gotten better over time, but all models produce a lack of flux (over opacity) in the K-band. M8V 1997 1995 2001 Allard et al. 2010 1971 Dust Clouds The cloud composition according to equilibrium chemistry changes from: Zirconium oxide (ZrO2) Perovskite and corundum (CaTiO3, Al2O3) Silicates: forsterite (Mg2SiO4) Salts: (CsCl, RbCl, NaCl) Ices: (H2O, NH3, NH4SH) M → L → T dwarfs At Teff < 2200 K the cloud layers become optically thick enough to initiate cloud convection. Intensity variation due cloud formation and granulation. Teff = 2600 K : No dust formation Teff = 2200 K : dust has maximum optical thick density Teff = 1500 K: Dust starts to settle and gravity waves causing regions of condenstation Allard et al. 2010 Of course these models do not include rotation and Brown Dwarfs can have high rotation rates. Jupiter is as a good approximation as to what a brown dwarf atmosphere really looks like Teff = 2900 K, log g=5-0 model compared to GJ 866 Infrared 3mm 8000 Å Most cool star models have a use a more complete line lists for molecules, including water, and also include dust in the atmosphere Optical 4000 Å 10000 Å Discrepancies are due to missing opacities Comparison of Models NextGen: overestimating Teff Ames-Cond/Dusty: underestimating Teff BT-Settl: Using Asplund Solar abundances Stars with „normal“ opacities Condensation Dust clouds Allard et al. 2010 Now days researchers just download models from webpages. Kurucz model atmospheres have become the „industry standard“, and are continually being improved. These are used mostly for stars down to M dwarfs. The Phoenix code is probably more reliable for cool objects. • Kurucz (1979) models - ApJ Supp. 40, 1 R.L. Kurucz homepage: http://kurucz.harvard.edu • The PHOENIX homepage P.H. Hauschildt: http://www.hs.uni-hamburg.de/EN/For/ThA/phoenix/index.html • Holweger & Müller 1974, Solar Physics, 39, 19 – Standard Model • Allens Astrophysical Quantities (Latest Edition by Cox) TEFF 5500. GRAVITY 0.50000 LTE ITLE SDSC GRID [+0.0] VTURB 0.0 KM/S L/H 1.25 OPACITY IFOP 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 0 0 0 CONVECTION ON 1.25 TURBULENCE OFF 0.00 0.00 0.00 0.00 BUNDANCE SCALE 1.00000 ABUNDANCE CHANGE 1 0.91100 2 0.08900 ABUNDANCE CHANGE 3 -10.88 4 -10.89 5 -9.44 6 -3.48 7 -3.99 8 -3.11 ABUNDANCE CHANGE 9 -7.48 10 -3.95 11 -5.71 12 -4.46 13 -5.57 14 -4.49 ABUNDANCE CHANGE 15 -6.59 16 -4.83 17 -6.54 18 -5.48 19 -6.82 20 -5.68 ABUNDANCE CHANGE 21 -8.94 22 -7.05 23 -8.04 24 -6.37 25 -6.65 26 -4.37 ABUNDANCE CHANGE 27 -7.12 28 -5.79 29 -7.83 30 -7.44 31 -9.16 32 -8.63 ABUNDANCE CHANGE 33 -9.67 34 -8.69 35 -9.41 36 -8.81 37 -9.44 38 -9.14 ABUNDANCE CHANGE 39 -9.80 40 -9.54 41 -10.62 42 -10.12 43 -20.00 44 -10.20 ABUNDANCE CHANGE 45 -10.92 46 -10.35 47 -11.10 48 -10.18 49 -10.58 50 -10.04 ABUNDANCE CHANGE 51 -11.04 52 -9.80 53 -10.53 54 -9.81 55 -10.92 56 -9.91 ABUNDANCE CHANGE 57 -10.82 58 -10.49 59 -11.33 60 -10.54 61 -20.00 62 -11.04 ABUNDANCE CHANGE 63 -11.53 64 -10.92 65 -11.94 66 -10.94 67 -11.78 68 -11.11 ABUNDANCE CHANGE 69 -12.04 70 -10.96 71 -11.28 72 -11.16 73 -11.91 74 -10.93 ABUNDANCE CHANGE 75 -11.77 76 -10.59 77 -10.69 78 -10.24 79 -11.03 80 -10.95 ABUNDANCE CHANGE 81 -11.14 82 -10.19 83 -11.33 84 -20.00 85 -20.00 86 -20.00 ABUNDANCE CHANGE 87 -20.00 88 -20.00 89 -20.00 90 -11.92 91 -20.00 92 -12.51 ABUNDANCE CHANGE 93 -20.00 94 -20.00 95 -20.00 96 -20.00 97 -20.00 98 -20.00 ABUNDANCE CHANGE 99 -20.00 EAD DECK6 72 RHOX,T,P,XNE,ABROSS,ACCRAD,VTURB 2.18893846E-03 2954.2 6.900E-03 1.814E+06 6.092E-05 1.015E-02 0.000E+00 0.000E+00 0.000E+00 2.91231236E-03 2989.5 9.180E-03 2.404E+06 6.205E-05 9.083E-03 0.000E+00 0.000E+00 0.000E+00 3.83968499E-03 3098.1 1.211E-02 3.187E+06 6.586E-05 5.843E-03 0.000E+00 0.000E+00 0.000E+00 5.04036827E-03 3099.5 1.590E-02 4.145E+06 6.588E-05 6.526E-03 0.000E+00 0.000E+00 0.000E+00 6.60114478E-03 3220.4 2.082E-02 5.367E+06 6.923E-05 4.064E-03 0.000E+00 0.000E+00 0.000E+00 8.62394743E-03 3221.5 2.722E-02 6.977E+06 6.963E-05 4.606E-03 0.000E+00 0.000E+00 0.000E+00 1.12697597E-02 3324.7 3.558E-02 8.957E+06 7.213E-05 3.199E-03 0.000E+00 0.000E+00 0.000E+00 1.47117179E-02 3325.9 4.649E-02 1.165E+07 7.271E-05 3.535E-03 0.000E+00 0.000E+00 0.000E+00 1.92498763E-02 3393.3 6.080E-02 1.503E+07 7.447E-05 2.979E-03 0.000E+00 0.000E+00 0.000E+00 2.51643328E-02 3425.0 7.950E-02 1.949E+07 7.579E-05 2.870E-03 0.000E+00 0.000E+00 0.000E+00 3.29103373E-02 3462.1 1.040E-01 2.526E+07 7.737E-05 2.733E-03 0.000E+00 0.000E+00 0.000E+00 4.30172811E-02 3502.2 1.359E-01 3.272E+07 7.906E-05 2.600E-03 0.000E+00 0.000E+00 0.000E+00 5.62014022E-02 3540.2 1.776E-01 4.238E+07 8.090E-05 2.473E-03 0.000E+00 0.000E+00 0.000E+00 7.33634752E-02 3576.1 2.318E-01 5.487E+07 8.294E-05 2.359E-03 0.000E+00 0.000E+00 0.000E+00 9.56649214E-02 3609.9 3.023E-01 7.100E+07 8.521E-05 2.267E-03 0.000E+00 0.000E+00 0.000E+00 1.24571186E-01 3643.2 3.936E-01 9.176E+07 8.778E-05 2.219E-03 0.000E+00 0.000E+00 0.000E+00 1.61931251E-01 3676.7 5.117E-01 1.184E+08 9.071E-05 2.191E-03 0.000E+00 0.000E+00 0.000E+00 2.10049977E-01 3710.3 6.637E-01 1.525E+08 9.409E-05 2.159E-03 0.000E+00 0.000E+00 0.000E+00 2.71775682E-01 3744.2 8.588E-01 1.959E+08 9.802E-05 2.123E-03 0.000E+00 0.000E+00 0.000E+00 3.50602645E-01 3779.0 1.108E+00 2.511E+08 1.026E-04 2.087E-03 0.000E+00 0.000E+00 0.000E+00 4.50760286E-01 3814.6 1.424E+00 3.209E+08 1.080E-04 2.071E-03 0.000E+00 0.000E+00 0.000E+00 5.77317822E-01 3851.4 1.824E+00 4.086E+08 1.142E-04 2.062E-03 0.000E+00 0.000E+00 0.000E+00 7.36336604E-01 3889.2 2.327E+00 5.186E+08 1.216E-04 2.038E-03 0.000E+00 0.000E+00 0.000 Sample grid from Kurucz