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Modelling Stellar Spectrospheres

Kenneth R Whight

Abstract A relatively simple model of stellar spectrospheres is presented and calibrated on the known

measured properties of the Solar spectrosphere. The software that implements the model is freely

available and can be used by amateur astro-spectroscopists to estimate properties of stellar

spectrospheres such as temperature, pressure, density and thickness.

1.0 Introduction For the purposes of this work a "spectrosphere" is defined as that region of a star's atmosphere in

which an absorption line series is formed e.g. the hydrogen Balmer series. Spectrospheres vary in

position within a star's atmosphere according to the particular element and the state of ionisation

associated with a particular line series.

High resolution (R~18000) stellar spectral lines that can be measured using amateur equipment

contain, in their line profiles, information regarding the physical properties of the star's

spectrospheres, among those properties are temperature, pressure, density and thickness. In reality

all but the last of these properties are functions of depth and are the subject of complex models

developed by professional astrophysicists.

In a previous paper1 I developed a simple model of a spectrosphere, I will now describe an extension

to the analysis that allows properties of particular spectrospheres to be estimated e.g. average

pressure, average mass density and effective thickness. The complete model has been implemented

in custom software so that measured Hydrogen Balmer absorption lines can be analysed and model

parameters determined for a specific star. The temperature and mean-free-path (MFP) between

particle collisions can be estimated purely from the overall shape of the continuum and detailed

absorption line profiles respectively but to complete the process of estimating the spectrosphere

properties for an arbitrary star the model must first be calibrated on known properties of the Sun.

In this paper I will first analyse the Sun's Hline to obtain the data necessary to complete the current

aims and then go on to extend the theory developed in reference 1. I shall then apply the theory to a

selection of star's to obtain estimates of properties of their spectrospheres using my custom

software.

The custom software I have developed to implement this model reads and writes data file formats

used by RSpec2. The software will also work with other packages provided they can present spectral

data in two column data format but in this case model parameters will not be automatically

displayed in the spectral graphs, rather they will be stored in a separate data file.

The model assumes that a star's spectrosphere is comprised of a single layer in thermodynamic

equilibrium and as such it has a single value for temperature, pressure and density independent of

depth. The parameters of the model are:-

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Figure 1: H Europa solar reflection line

AMass: the effective atomic mass used in the calculations.

L0: the line centre wavelength.

dl: the wavelength bin width (uniform over the spectrum)

T: the Temperature.

MFP: the atomic mean free path.

vrot: the equatorial surface velocity of rotation.

Ob: the oblateness i.e. equatorial divided by polar diameter.

Theta: the angle of view relative to the equatorial plane.

In the case of stars that are known to be fast rotators, speed of rotation, oblateness and viewing

angle relative to the equator are relevant parameters but as rotation is not a factor in solar spectra

this will not be discussed further here.

For the purpose of modelling spectral line profiles and deriving a pressure value from the particle

mean free path, the spectrosphere is assumed to be comprised of a single "average" atomic species

"Hyelium" with an effective atomic mass = 1.255 (determined by H and He cosmic abundances i.e.

91.5% H and 8.5% He atoms by number) a different effective mass can be input to simulations if

desired.

Real stars will deviate from the predictions of this simple model according to how well they are

approximated by the model assumptions however, knowing how a single layer single atomic species

thermal equilibrium spectrosphere should look allows speculation as to the cause of inevitable

departures from the model.

2.0 Measurement and Analysis of the Sun's HLine The equipment used to capture experimental high resolution spectra for this work comprised a

Shelyak LhiresIII spectrograph with 2400 line/mm grating mounted on a 12" Meade LX200 telescope.

Rather than use low resolution spectra to obtain an estimate for the Sun's temperature the

temperature was simply assumed to be that of a typical G2v type star i.e. 6000K.

A high resolution (R~18000) spectra at the H

wavelength 6563 Angstrom was captured by

reflection from Jupiter's moon Europa. It was

necessary to use reflected light from Europa as

the analysis requires an accurate zero level for

the spectral intensity. To achieve this the

spectral reduction process, using Rspec,

involved subtracting a nearby stripe of dark sky

background from the spectral stripe image of

Europa, both stripes being of equal width. It

does not matter if the reflectivity of Europa

varies between the two wavelengths we only

require and assume that it is constant across

each spectral line. Figures 1 shows the

measured, calibrated and background subtracted Europa absorption line at H.

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When my software (Profile.exe) is run with a measured, wavelength calibrated, line profile as input

the it first re-samples the spectrum to a uniform wavelength bin width and then removes any linear

continuum component before outputing the normalised (continuum = 1) "flattened" absorption line.

It also outputs the effective emission line that is to be used as a target reference for the iterative

modelling process, figures 2 and 3 show the outputs that correspond to the input spectrum

displayed in figure 1.

Relevant model parameters are displayed

automatically in these figures, those parameters being:-

L0 the line centre wavelength in Angstrom

dL the re-sampled wavelength bin width in Angstrom

A0 the intensity of the measured absorption line at the central wavelength

The equivalent width (area) of the line.

To continue the modelling process the

custom software is again run but this

time using the effective emission line

reference spectrum as input (figure 3).

The required atomic mass, simulation

temperature and mean free path are

requested and a corresponding

convolved emission line profile is

output. Figure 4 shows the final "best

fit" achieved at the end of a manual

iterative process and the parameters

again automatically appear in the

spectra displayed by Rspec. It can be

seen that there is a compromise to be

made between fitting the central core

of the line and the outer "wings". The

final, line profile, model parameter

Figure 4: Modelled effective H emission line(red)

measured (blue)

Figure 2: Flattened normalised H line

Figure 3: H effective emission line

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values are:

Effective particle atomic mass Amass = 1.25

Temperature T= 6000K

Mean free path MFP = 467A

Equation (2.15) of reference 1 can

now be used to generate the

corresponding modelled

absorption line as we know

. This is achieved

by running the custom software

again but now using the modelled

emission line as input, the result is

displayed in figure 5.

In this figure we can see that a

new parameter is displayed

namely sN2t. This parameter is

proportional to the column

density of neutral atoms in

principle quantum state n = 2.

In the following sections we will further develop the analysis given in reference 1 and show how this

modelling data can be used to estimate interesting properties of a star's spectrosphere.

3.0 Theory The known properties of the Sun's spectrosphere will be used as a calibration to estimate additional

properties of other stellar spectrospheres. First we need to determine and calibrate a relationship

between mean free path and pressure. Once this is achieved we can go on to derive and calibrate an

expression for the layer thickness of a star's spectrosphere.

3.1 Relationship Between Mean-Free-Path and Pressure We know that the concentration of neutral atoms is the reciprocal of the average volume swept out

by a neutral atom between collisions i.e:-

(3.1)

where is the number density of neutral atoms in a spectrosphere, l is the mean free path

(determined from modelling a Balmer line profile) and the "effective" atomic collision cross sectional

area .

The Saha equation states for Hydrogen that:-

(3.2)

Figure 5: Modelled H absorption line(red) measured (blue)

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where is the ionised hydrogen concentration and is the electron thermal de Broglie

wavelength

. We also have:-

(3.3)

Given that can be determined for any star from the analysis of spectral line profiles1 then, if we can

determine a value for the "effective" radius of an atom , we should be able to estimate .

Splitting the atomic collision cross-section into a neutral-neutral and neutral-ionized atom part we

can write a weighted average for :-

(3.4)

where is the "effective" radius of a neutral atom and is the "effective" radius of an ionized

atom. In addition, as quantum mechanically an electron wave function has a significant value a

considerable distance beyond the classical radius, a multiplier (qm) has been introduced. It is also

assumed that is independent of temperature.

We can calculate a weighted average classical radius for a neutral atom at any given temperature

but to calculate we have to calibrate on the known properties of the Sun. For the Sun we know

both the "mean free path" from measurement (Section 2.0) and the "total atomic

number density" from published data so we should be able to determine .

Equations (3.2) and (3.3) can be combined to obtain a quadratic in and applied to the Sun we

have:-

(3.5)

On solving equation (3.5) for we can then determine from equation (3.1) and from

equation (3.2), which then enables equation (3.4) to be applied to obtain a value for with the

result that 5.489e-7 m.

Now for an arbitrary star we can substitute equation (3.4) into equation (3.1) to obtain:-

(3.6)

which together with equation (3.2) forms a quartic equation in i.e.:-

(3.7)

Where

,

and

Equation (3.7) is best solved numerically for and then, having obtained by squaring, we can

determine from equation (3.2) and finally from equation (3.3).

We can now go on to determine the pressure of a spectrosphere produced by the neutral atoms, the

perfect gas law states that:-

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Figure 6: Feynman diagram for atomic

photon absorption

(3.8)

where P is pressure, V is volume, T is absolute temperature, n is the number of moles of the

particles, R (= 8.31441) is the molar gas constant therefore:-

(3.9)

where is the number of moles of the particles per unit volume, but is the number of neutral

particles per unit volume (that we now know) so:-

(3.10)

where is Avogadro's number (= 6.022045e23). An alternative way of writing the same equation

is:-

(3.11)

Where k is Boltzmann's constant (=1.380662e-23). Which for the Sun yields:-

(3.12)

3.2 Estimating the layer thickness of a spectrosphere From an analysis of spectral line shapes1 we can obtain a value proportional to the

column density of neutral atoms in the n=2 principle quantum state i.e:-

(3.13)

where is the effective thickness of the spectrosphere and is a "yet to be determined" constant of

proportionality. We can also write:-

(3.14)

where A is the Lyman wavelength. Combining these two equations we arrive at the

following expression for :-

(3.15)

We need to determine a value for the constant before we can determine a value for .

To do this we could calibrate on the properties of

the Sun however we should be able to appeal to

fundamental physics to determine a value for .

The fundamental physics involved is Quantum

Electrodynamics and as the absorption of a single

photon by a single electron is kinematically

forbidden3 figure 6 shows the simplest atomic

absorption process. It involves a photon exchange

with the nucleus as described by this 3 vertex

Z

e-

e-

Z e-

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Figure 7: Modelled H absorption line and

calculated spectrosphere parameters

Feynman diagram. In this diagram the atomic nucleus is labelled Z, the electron is labelled e- and the

photons are indicated by wiggly lines. It can be seen that a nucleus, electron and photon enter from

the left whilst just the nucleus and an excited electron exit to the right. Three vertices imply that the

differential interaction cross section includes a dimensionless factor of

where is the fine structure constant. Integration over all angles introduces a further factor of

yielding:-

(3.16)

This value is used in my custom software to estimate the thickness of stellar spectrospheres and

yields a result for the Sun's spectrosphere thickness that is in good agreement with the known value

of approximately 400km which is the thickness of the Sun's photosphere.

3.3 Software implementation An additional program SpecSph.exe has been written to implement the theory just described. When

run on the modelled data, as displayed in

figure 5, it calculates properties of a star's

spectrosphere and those properties are

again displayed automatically in the Rspec

generated graph as depicted in figure 7.

SpecSph.exe has been calibrated on the Sun

so it is not surprising that the Sun's known

properties are accurately reproduced.

The additional properties calculated and

displayed by the software are:-

Pressure P = 1.010e-1 Bar

Density Rho = 2.531e-4 kg/m3

Spectrosphere thickness tss =

4.705e5 m

Neutral atom radius rn = 2.646e-110

m

Neutral atom radius multiplier qm =

5.0

Ionised atom radius ri = 5.489e-7 m

Ionised atom concentration NII =

2.259e19 m-3

Neutral atom concentration NI =

1.219e23 m-3

Table 1 displays published temperature

and pressure profiles as a function of

depth for the Sun. It can be seen that

the calculated pressure at 1.01e-1 Bars is

a reasonable average value.

Table 1: Published data on the Solar photosphere

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Figure 8: AlbireoB: Modelled H absorption line and

calculated spectrosphere parameters

Figure 9: Deneb: Modelled H absorption line and

calculated spectrosphere parameters

4.0 Analysis of Other Stars A number of other star's have been fully analysed using my custom software, the results of the

spectrosphere analysis will be given in this section.

4.1 AlbireoB (Cygni)

The B8V type B component of this

double star system, officially

designated TYC 2133-2963-1, was

modelled using the measured H

absorption line as a template. The

final result of running my analysis

programs on this star is depicted in

figure 8.

The key spectrosphere parameter

value estimates are:-

Pressure: P = 2.117e-7 Bar

Density: Rho = 5.223e-7

kg/m^3

Thickness: tss = 3.368e6 m

Neutral atom density: NI = 8.700e16 m^-3

Ionised atom density: NII = 2.515e20 m^-3

4.2 Deneb (Cygni) The type A2 1a star Deneb was

modelled using the measured H

absorption line as a template. The

final result of running my analysis

programs on this star is depicted in

figure 9.

The key spectrosphere parameter

value estimates are:-

Pressure: P = 3.018e-6 Bar

Density: Rho = 1.433e-7

kg/m^3

Thickness: tss = 2.224e6 m

Neutral atom density: NI =

1.943e18 m^-3

Ionised atom density: NII =

6.707e19 m^-3

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Figure 10 : Sulaphat: Modelled H absorption line and

calculated spectrosphere parameters

4.3 Sulaphat (Lyr)

The type B9III star Sulaphat was modelled using measured H absorption line as a template. The final

result of running my analysis programs on this star is depicted in figure 10.

The key spectrosphere parameter value estimates are:-

Pressure: P = 6.845e-8 Bar

Density: Rho = 3.298e-7

kg/m^3

Thickness: tss = 1.820e7 m

Neutral atom density: NI =

2.750e16 m^-3

Ionised atom density: NII =

1.589e20 m^-3

5.0 Discussion and Conclusions This paper follows on from my previous work (reference 1), which is embodied in my analysis

program Profile.exe, in that paper I was confident of the underlying physics and followed the

mathematics to its inevitable end. The predictions can also be tested and were, allowing for the

limitations of the model, reasonably accurate.

In this work, which is embodied in my analysis program SpecSph.exe, I am less familiar with the

physics and so the results are necessarily more speculative particularly as they cannot be tested on

our nearest star as that star (the Sun) was used to calibrate the model.. That being the case I would

welcome criticism (I can be contacted via the RSpec yahoo usergroup) that increases my knowledge

and would allow me to improve/correct the analysis.

I would to have been able to calculated the neutral to ionised atom collision cross section rather

inferred it from properties of the Sun, the assumed linear extrapolation with ionised atom

concentration could lead to significant errors as analysed stars become increasingly dissimilar from

the Sun. I have not managed to find any data on the predictions of professional models regarding

the average properties of the stars modelled here and so do not know how safe my extrapolations

are.

In conclusion I have fully developed a "simple" model of a star's spectrosphere that allows me to

extract as much information as possible from my own amateur measurements. Hopefully this work

and the accompanying software, which is freely available from my website1, is of interest to other

amateur spectroscopists.

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References 1. "Modelling Stellar Absorption Lines": www.thewhightstuff.co.uk

2. www.rspec-astro.com

3. "Special Relativity From Einstein to Strings", Patricia M. Schwarz and John H. Schwarz,

Cambridge University Press, 2004, ISBN 0 521 81260 7.