- 83 Views
- Uploaded on
- Presentation posted in: General

Model Photospheres

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Model Photospheres

- What is a photosphere?
- Hydrostatic Equilibruium
- Temperature Distribution in the Photosphere
- The Pg-Pe-T relationship
- Properties of the Models
- 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:
- Spherical
- Can pulsate
- Granulation, starspots, velocity fields
- Magnetic fields
- Winds and mass loss

- Our model:
- Plane parallel geometry
- Hydrostatic equilibrium and no mass loss
- Granulation, spots, and velocity fields are represented by mean values
- No magnetic fields

P +dP

A

dr

dm

P

Gravity

II. Hydrostatic equilibrium

dA

P + dP

r + dr

P

r

M(r)

The gravity in a thin shell should be balanced by the outward gas pressure in the cell

Fp = PdA (P + dP)dA = dP dA

Pressure Force

dM

FG = GM(r)

Gravitational Force

r2

P +dP

A

r

dr

M(r) = r(r) 4pr2dr

0

dm

dM = r dA dr

P

Gravity

FP + FG = 0

Both forces must balance:

r(r)M(r)

G

dP dA

+

dA

dr

= 0

r2

=

- Radiation pressure:

2.52 1015

T4

dyne/cm2

2. Magnetic pressure:

PM =

B2

4s

T4

PR =

8p

3c

3. Turbulent pressure:

~ rv2v is the root mean square velocity of turbulent elements

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:

B2

8p

Footnote: Magnetic pressure is what is behind the emergence of magnetic flux ropes in the Sun

Pphot

Ptube

+

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

B is for magnetic pressure = Pg

v is velocity that generates pressure equal to Pg according to rv2

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.

r(r)M(r)

G

=

dP

dr

r2

dP = grdx

The weight in the narrow column is just the density volume gravity

g

dP

kn

=

dtn

In our atmosphere

F

GM/r2 = g (acceleration of gravity)

x increases inward so no negative sign

P

dA

dx

r

F + dF

P + dP

dtn = knrdx

gravity

Pg dPg = Pg

dk0

k0 is a reference wavelength (5000 )

g

3

2

3

2

2

3

k0

3/2

dt0

Pg(t0) =

Pg

gPg

to

to

(

dt0

(

Pg(t0) =

g

g

k0

k0

0

0

t0

log to

Pg

(

dlog t0

(

Pg(t0) =

k0 log e

One way to integrate the hydrostatic equation

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.

Limb darkening is due to the decrease of the continuum source function outwards

In(0) =

Snetnsec q sec q dtn

0

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.

III. Temperature Distribution in the Solar Photosphere

- Two probes of depth:
- Limb Darkening
- Wavelength dependence of the absorption coefficient

The increment of path length along the line of sight is ds = dx sec q

q

ds

Temperature profile of photosphere

Bottom of photosphere

10000

8000

z

Temperature

6000

4000

q2

z=0

q1

dz

tn =1 surface

Top of photosphere

Limb Darkening

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

z

Temperature profile of photosphere and chromosphere

10000

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

8000

chromosophere

Temperature

6000

4000

z=0

At 1.3 mm the solar atmosphere exhibits limb brightening

Horne et al. 1981

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

To get an accurate measurement of the planet radius you need to model the limb darkening appropriately

q

If you define the radius at which the intensity is 0.9 the full intensity:

At l=10000 , cos q=0.6, q=67o, projected disk radius = sin q = 0.91

At l=4000 , 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

Using:

a + b cosq

In(0) =

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

In(0) =

Snetnsec q sec q dtn

0

In the grey case had a linear source function:

Sn = a + btn

Limb darkening laws usually of the form:

Ic(0) continuum intensity at disk center

Ic = Ic(0) (1 e + e cos q)

e 0.6 for the solar case, 0 for A-type stars

d log tn

log e

Rewriting on a log scale:

In(0) =

Snetnsec qtn sec q

0

Contribution function

Sample solar contribution functions

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.

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 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

[t + q(t)]

Teff

T(t) =

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(t0)

In the grey case:

Teff

In the grey case the scaling factor is the ratio of effective temperatures

T(t) =

T(t)

Teff

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

Fj(T)

N1j

=

Pe

N0j

u1

u0

IV. The PgPeT 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 = number of ions per unit volume of the jth element

See Saha equation from 2nd lecture

N0j =number of neutrals

5/2

F(T) = 0.65

T

105040I/kT

Nej

Nej

=

=

Fj(T)

N0j

Nj Nej

Pe

Fj(T)/Pe

Nej

=

Nj

1 + Fj(T)/Pe

Neglect double ionization. N1j = Nej, the number of electrons per unit volume that are contributed by the jth element.

The total number of jth element particles is Nj = N1j + N0j. Solving for Nej

The pressures are:

Pe = SNejkT

j

Pg = S (Nej +Nj)kT

j

SNejkT

Pe

=

S (Nej +Nj)kT

Pg

Fj(T)/Pe

Using the number abundance Aj = Nj/NH

NH = number of hydrogen

SNj

Pe

1 + Fj(T)/Pe

=

Pg

Fj(T)/Pe

SNj

1 +

1 + Fj(T)/Pe

Fj(T)/Pe

SAj

1 + Fj(T)/Pe

Pg

Pe

=

Fj(T)/Pe

SAj

1 +

1 + Fj(T)/Pe

Taking ratios:

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).

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

log t

1

0

1

2

The absorption coefficient kn is largely due to the negative hydrogen ion which is proportional to Peso the opacity increases very rapidly with depth.

3

4

F(T)/Pe

Pe = Pg

1 + 2 F(T)/Pe

2

Pe = F(T)Pg 2F(T)Pe = F(T)(Pg 2Pe)

2

Pe F(T)Pg

Hydrogen dominates at high temperatures and when it is fully ionized Pg 2Pe

At cooler temperatures Pe ~ Pg

Where does the later come from? Assume the photosphere is made of single element this simplifies things:

Pg >> Pe in cool stars

3

2

g

t0

log to

Pg

(

dlog t0

(

Pg(t0) =

k0 log e

Completing the model

- 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

t0

1

dt0

x(t0) =

k0(t0)r(t0)

0

PgPe

NNe

NH =

=

kT S Aj

S Aj

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

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

log t0

S AjkT(t0)t0

x(t0) =

k0(t0)S Ajmj[Pg(t0)t0 Pe(t0)]

A more interesting form is to integrate on a Pg scale with dPg = rgdx

d log t0

d log e

Pg

S AjkT(p)

x(t0) =

S Ajmj

1

g

dp

This makes physical sense if you recall the scale height of the atmosphere:

Scale height H = kT/mg

p

The thickness of the atmosphere is inversely proportional to the surface gravity since T(Pg) depends weakly on gravity

It is customary to integrate on a log t scale

0

kn(t0)t0 d log t0

Fn(0)= 2p

Sn(tn)E2(tn)

k0(t0) d log e

Flux contribution function

Computation of the Spectrum

The spectrum

Fn(0)= 2p

Sn(tn)E2(tn)dtn

Flux Contribution Functions as a Function of Wavelength

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 at 8000 originates higher up in the atmosphere than flux at 5000 or 3646

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

d Sn(tn)

E3(tn)dtn

Fn(0)= pSn(0) +

0

dtn

0

The flux arises from the gradient of the source function. Depths where dS/dt is larger contribute more to the flux

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

Cannot scale T(Pg), unlike T(t0)!!!

V. Properties of Models: Pressure

d log T/dlog Pg = 0.4

Convection gradient

Temperature

Relationship between pressure and temperature for models of effective temperatures 3500 to 50000 K. The dashed line marks where the slope exceeds 11/g 0.4 and implies instability to convection

=

=

0.85

0.62

dlogPg

dlogPg

dlog g

dlog g

Teff = 8750 K

Effects of gravity

Increasing the gravity increases all pressures. For a given T the pressure increases with gravity

3

2

g

t0

log to

Pg

(

dlog t0

(

Pg(t0) =

k0 log e

Pg C(T) g 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 Pg in cool stars Pe constant g

Pe 0.5 Pg in hot stars Pe constant g

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

Pe

Pe

Pe

NH

kT

kT

Pg

Pg

Pg

S N j

=

Pe

PH

S N jkT

S N jkT

Qualitatively:

Using SAj for the sum of the metal abundances

1

=

S A j

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

g

1

1

S N 1j

Pe

Pe

k0

S A j

S A j

S (N 1 + N0)j

Pg

Pg

1

g

dk0

g

=

=

Pek0/Pe

PgSAjk0/Pe

dk0

dPg =

In the solar case metals are ionized SN1j >> N0j

k0 is dominated by the negative hydrogen ion, so k0/Pe is independent of Pe

t0

1

g

2

dt0

Pg

=

k0/Pe

S Aj

g and T are constants

0

For metals being neutral: SN1J << S(N1 + N0)j can show

Pe =c0

Pg =c0

Pg =c0

Pe =c0

(S Aj)

(S Aj)

(S Aj)

(S Aj)

Integrating:

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)

The mapping between optical depth and a real depth

Depth (km)

Log t

Why do you need to know the geometric depth?

Amplitude (mmag)

Wavelength (Ang)

In the case the pulsating roAp stars, you want to know where the high amplitude originates

Pe Pg

Pgas

Log (Pressure)

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:

- Opacities at low temperatures (molecules, incomplete line lists etc.) not well known

- Convection much stronger (fully convective)

- Condensation starts to occur (energy of condensation, opacity changes)

- Formation of dust

- Chemical reactions (in hotter stars the only reactions are ionization which is give by ionization equilibrium)

1971

2001

1995

1997

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

Allard et al. 2010

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

Most cool star models have a use a more complete line lists for molecules, including water, and also include dust in the atmosphere

Infrared

3mm

8000

Optical

Discrepancies are due to missing opacities

10000

4000

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 & Mller 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