A new modelling approachA new modelling approachfor DACs and SACs regionsfor DACs and SACs regionsin the atmospheres of hot in the atmospheres of hot

emission starsemission stars*Danezis E., *Lyratzi E, *Antoniou A.Danezis E., *Lyratzi E, *Antoniou A.,, **Popovi **Popovićć

L. L. ČČ., **Dimitrievi., **Dimitrievićć M. S. M. S.

*University of Athens, School of Physics, Department of *University of Athens, School of Physics, Department of Astrophysics, Astronomy and Mechanics, Astrophysics, Astronomy and Mechanics,

Panepistimiopolis, Zografos 157 84,Panepistimiopolis, Zografos 157 84,Athens – GreeceAthens – Greece

**** Astronomical Observatory, Volgina 7, 11160 Astronomical Observatory, Volgina 7, 11160 Belgrade – SerbiaBelgrade – Serbia

The The GRGR model modelOne of the main hypotheses when we constructed an old One of the main hypotheses when we constructed an old version of our model version of our model (rotation model),(rotation model), was that the line’s width was that the line’s width is only a rotational effectis only a rotational effect and we considered and we considered spherical spherical symmetrysymmetry for the independent density regions. for the independent density regions.

In a In a new approachnew approach of the problem we also consider of the problem we also consider the the random velocitiesrandom velocities in the calculation of the distribution function in the calculation of the distribution function L that we can detect in the line functionL that we can detect in the line function..

This newThis new LL is a synthesis of the is a synthesis of the rotational distribution Lrrotational distribution Lr that that we had presented in the old rotational model we had presented in the old rotational model and a Gaussianand a Gaussian that well defines the random velocities. This means that the that well defines the random velocities. This means that the new L has two limits, the first one gives us a Gaussian and the new L has two limits, the first one gives us a Gaussian and the other the old rotation distribution Lr. other the old rotation distribution Lr.

ggg

jejejej

iii LLSLII expexp1exp0

Ai (λlab)Vrad

V

φ

φ

equator

The new calculation of the distribution functions L

Let us consider a spherical shell and a point Ai in its equator. If the laboratory wavelength of a

spectral line that arises from Ai is λlab, the observed wavelength will be λ0=λlab+Δλrad

Danezis, E., Lyratzi, E., Nikolaidis, D., Antoniou, A., Popović, L. Č. & Dimitrijević, M. S., 2007 PASJ, 59, 4

If the spherical density region rotates, we will observe a displacement Δλrot and the new wavelength of the center of the line λi

is: where

where is the observed rotational velocity of the point Ai.

This means that

and if then

roti 0 sin0zrot

sin0

rotrot

c

Vz

rotV

sin1sin 000 zzi

22

sin10 zi

Ai (λ0)Vrot

φ

φ

The calculation of the distribution functions L

If we consider that the spectral line profile is a Gaussian,is a Gaussian, then we have: 

 

where κκ is the mean value of the distribution and in the is the mean value of the distribution and in the case of the line profile it indicates the center of the case of the line profile it indicates the center of the spectral line that arises from Aspectral line that arises from A

ii..

This means that:

2

2

2

1

eP

2

20

20

2

sin1

2

sin1

2

1

2

1

zz

eeP

For all the semi-equator we have

 

If we make the transformation ,

and

the above function (4) will be transformed and finally we have the function (5) :

)4.......(cos2

12

2

2

sin12

20

deLz

xsin

2

10 zx

u

2

)1(

2

)1(0

0

0

21)(

z

z

u duez

L

)5...(1

)(2

)1(

0

2

)1(

00

0 0

22

z z

uu dueduez

L

The above integrals have the form of a known integral erf(x)The above integrals have the form of a known integral erf(x)

that has the following propertiesthat has the following properties: :

x

u duexerf0

22

1.               

2.               

3.                as

4.               

5.               

6.               

00 erf

1erf 1lim

erfx

11 erf

...

!37!25!13

2 753 xxxxxerf

...2

531

2

31

2

111

32222

2

xxxx

exerf

x

xerfxerf

(6) 

The distribution function from the semi-spherical region is: 

(7)

(Method Simpson)

2

1

2

1

200

0

zerf

zerf

zL

2

2

0000

0

coscos22

cos222

d

zerf

zerf

zL final

This LThis Lfinalfinal((λ)λ) is the distribution that replaces the old is the distribution that replaces the old

rotational distribution L that rotational distribution L that our group proposed some years ago (Danezis et al 2001).our group proposed some years ago (Danezis et al 2001).

In the proposed distribution an important factor

is  This factor indicates the kind of the distribution that fits the line profile.

20 zm

3m1. If we have a mixed distribution. The line broadening is an effect of two equal reasons:

Discussion

a. The rotational velocity of a. The rotational velocity of the spherical region and the spherical region and

b. The random velocities of the ionsb. The random velocities of the ions ..

500m2.2. If m If m ≈≈ 500 500 the line broadening is only an effect of the the line broadening is only an effect of the

rotational velocityrotational velocity and the random velocities are very low. and the random velocities are very low. In this case In this case the profile of the line is the same with the profile of the line is the same with

the profile that we can produce using the old rotation modelthe profile that we can produce using the old rotation model (Danezis 2001, 2003). (Danezis 2001, 2003).

500m

3.3. Finally, if Finally, if m<1m<1 the line broadening is only an the line broadening is only an effect of random velocitieseffect of random velocities

and the line distribution and the line distribution is a Gaussianis a Gaussian..

An important point of our study is the calculation of the column densitycolumn density from our modelfrom our model. Lets start from the definition of the optical depth:

where ττ is the optical depth (no units),

kk is the absorption coefficient ( ),

ρρ is the density of the absorbing region ( ),

ss is the geometrical depth (cm)

s

dsk0

gr

cm2

3cm

gr

The column densityThe column density

Danezis, E., Lyratzi, E., Nikolaidis, D., Antoniou, A., Popović, L. Č. & Dimitrijević, M. S., 2007 PASJ, 59, 4

In the model we set , so

where LL is the distribution function of the absorption coefficient k k and has no units,ΩΩ equals 11 and has the units of kk ( )

We consider that for the moment of the observation and for a significant ion, k k is constant, so k k (and thus LL and ΩΩ) may come out of the integral. So:

We set

and τ τ becomes

Lk s

dsL0

gr

cm2

1

s

dsL0

s

ds0

L

For every one of ξ ξ along the spectral line (henceforth called ξξii)

we have that: 

We set

As contributes only to the units, σσii takes the value

of ξξii .

s

is

i

s

i dsdsds000

s

ii ds

0

)1.(2

gr

cm

Absorption lines

For each of λλii along the spectral line, we extract a σσii from each ξξi. i. The program we use calculates the ξξii for the centre of the line. This means that from this ξξii we can measure the respective σσii .

If we add the values of all σσii along the spectral line then we

have

( in ),

which is the surface densitysurface density of the absorbing matter, which creates the spectral line.  If we divide σ σ with the atomic weightatomic weight of the ion which creates the spectral line, we extract the number density of the number density of the absorbers,absorbers, meaning the number of the absorbers per square centimetre

( (in )).  

i

i2cm

gr

AWn

2cm

This number density corresponds to the energy densitythe energy density which is absorbed by the whole matter which creates the observed spectral line ( ( in )) and which is calculated by the is calculated by the model. model.  It is well known, that each absorber absorbs the specific amount of the energy needed for the transition which creates the specific line. This means that if we divide the calculated energy densitywe divide the calculated energy density ( ) with the energy needed for the transition, wewith the energy needed for the transition, we

obtainobtain the column densitythe column density (in ). 

AW

E2cm

erg

AW

E

2cm

 In the case of the emission lines we have to take into account not only ξe, but also the source function S, as both of these parameters

contribute to the height of the emission lines. So in this case we have:

where: j j is the emission coefficient ( ),

kk is the absorption coefficient ( )

ρρee is the density of the emitting region ( )

ss is the geometrical depth (cm)

s

ee dsk

jS

0

Aradsgr

erg

gr

cm2

3cm

gr

Emission lines

We set

where L L is the distribution function of the absorption coefficient kk and has no units,Ω Ω equals 11 and has the units of k k ( )

And

where LLee is the distribution function of the emission coefficient jj

and has no units,

ΩΩee equals 11 and has the units of jj

As we did before, in the case of the absorption lines, we may consider that ΩΩ may come out of the integral.

Lk

gr

cm2

1

eeLj

)1(Aradsgr

erge

So: As in the model we use the same distribution for the absorption and for the emission, .

So:

We set

As contributes only to the units,

σσee takes the value of SSξξe.e.

s

eee

s

eee

s

eee

s

ee dsL

Lds

L

Lds

L

Lds

k

jS

0000

LLe

s

ee

es

eee dsS

dsS00

s

ee

ee ds

S

0

)1(Aradsgr

erge

For each λFor each λii along the spectral line, we extract a along the spectral line, we extract a σσii from each S from each Sξξee..

The program we use calculates the ξξee for the center of the line and

the SS. This means that from this ξξee and SS we can measure the

respective σσii..

 If we add the values of all σi along the spectral line then we have

(in ),

which is the surface densitysurface density of the emitting matter, which creates the spectral line. If we divide σ with the atomic weight of the ion which creates the spectral line, we extract the number densitythe number density of the emitters, meaning the number of the emitters per square centimetre

(in ).

i

i2cm

gr

AWn

2cm

This number densitynumber density corresponds to the energy densityenergy density which is emitted by the whole matter which creates the observed spectral line

( (in )) and which is calculated by the model.is calculated by the model.

It is well known, that each emitter emits the specific amount of the energy needed for the transition which creates the specific line.

This means that if we divide the calculated energy density( )

with the energy needed for the transitionenergy needed for the transition, we obtain the column column

densitydensity (in ).

AW

E2cm

erg

AW

E

2cm

The next presentation is about some important

remarks and applications of GR model

Thank you very much for your attention