Extinction Work

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[edit] Stellar Radiation

We have used Stellar Radiation Field Model,that is in Tauvex tools to calculate the flux of a star of any spectral types and luminosity classes.This calculation is based on Kurucz model(1992) and scaled down to visual magnitude which requires three parameters (Visual magnitude, extinction and spectral type) as input to obtain the spectrum of a star in a wavelength range of 900-10000\AA. The Kurucz model gives the surface flux for different combinations of temperature,gravity and metallicity of stars. The theory involved in this model is as follows:

If I0 is the intensity at the surface of a star , then the reduced intensity of the beam of the star after traveling trough dust clouds is

I = I0e − τ

but I = 4πd2f and I0 = 4πr2f0

where f=flux of a star at a distance d and r=radius of the star.

so, 4πd2f = 4πr2f0e − τ

f = (r / d)2f0e − τ

if f0 =Kurucz flux at 5500A = K(5500A)

f = (r / d)2K(5500A)e − τ

f = ΓK(5500A)e − τ

Γ = Scaling factor;

we know m-M = -2.5log(f/f0) and if we take vega star for comparison , then M=0. flux corresponding to zero magnitude f0= 3.64e-9 erg/cm^2/sec/A when m=v

so v=-2.5log(f/3.64e-9)

f=3.64e-9*10-v/2.5

\Gamma K(5500 \AA).e^{-\tau}=3.64\times10^{-9}\times 10^{-V/2.5} .....................eq(1)


τ = optical depth;

τ = Av / 1.0863 = E(B − V).Rv / 1.0863

E(B − V) = (B − V)O − (B − V)I

Extinction and visual magnitude are the unknown parameters along with flux K(5500) which is taken care of by the spectral type.Once we supply these parameters to stellar flux calculator means it'll give the spectrum corresponding to that spectral type.

[edit] Effective Wavelength

The effective wavelength is the wavelength of the homogeneous radiation, that under the same condition produce the same radiation in a beam of heterogeneous radiation.

The effective wavelength

\lambda_{eff} = \frac{\int \lambda F(\lambda) S(\lambda) d\lambda}{\int F(\lambda) S(\lambda) d\lambda}

For Galex

\lambda_{eff}= \frac{\int \lambda F(\lambda) A(\lambda) d\lambda}{\int F(\lambda) A(\lambda) d\lambda}


Where F(λ) = flux in erg/cm2/sec/A ,

A(λ) = Effective area corresponding to the wavelength λ

S(λ) =Filter response = 1

We have calculated the effective wavelength for Fuv and Nuv band of each spectral type using the following formula:

\lambda_{eff}= \frac{\sum \lambda F(\lambda) A(\lambda) \Delta \lambda}{\sum F(\lambda) A(\lambda) \Delta \lambda} .......................................eq(2)

We have assumed V=10 and E(B-V)=0.0 for all the spectral types to get the spectral energy distribution (λVsF(λ)) for them from the stellar flux calculator.Then separated out the spectrum in two ultraviolet bands (FUV:1350A-1750A and NUV:1750-2750A) and took corresponding effective area from galex pages for the two bands to obtain the effective wavelength for all spectral types using the above formula.The effective wavelength increases toward cooler stars and it varies with spectral type and, to a lesser extent, luminosity class. These are tabulated in table:1


[edit] Effective Area
The effective area for each of the two GALEX bands (A(λ)) is taken from the galex pages and their text files are given here as NUV band and the FUV band, respectively.The plot of the effective area or filter response is given in figure:1
Fig.1. Effective area for imaging mode.
Fig.1. Effective area for imaging mode.


[edit] Theoretical colour

Theoretical FUV and NUV fluxes are calculated in the same way as the effective wave length using the formula:

F(\lambda)= \frac{\sum F(\lambda) A(\lambda) \Delta \lambda}{\sum A(\lambda) \Delta \lambda} ..........................................eq(3)

Here also the stellar spectrum is obtained assuming visual magnitude,V=10 and extinction,E(B-V)=0.0,0.1,0.2 & so on , for all spectral types.The FUV and NUV fluxes are converted to their magnitudes(AB)after calculating them from the spectrum using the formula:

FUV: m(AB)=-2.5log10(F(λ)/1.40e-15)+18.82

Nuv: m(AB)= -2.5log10(F(λ)/2.06e-16)+20.08

Theoretical B-V is calculated using the reddening formula,

E(B − V) = (B − V)O − (B − V)I

where (B-V)I is calculated from the Zombeck, which is a classic book for several astronomical information.We have already assumed the E(B-V)to get the stellar spectrum.Hence we got the colours FUV-NUV and B-V for all spectral types and luminosity classes separately for different extinctions.

The colour of a star is dependent on the spectral types and and extinction but independent of the magnitude.The colour-colour plot of all spectral types for extinction=0.0(blue)and 0.2(red) is given in figure:2 .
Fig.2. colour-colour plot for E(B-V)=0.0(blue) and 0.2 (red).
Fig.2. colour-colour plot for E(B-V)=0.0(blue) and 0.2 (red).

table

[edit] Observed colour

A list of bright stars has been taken from the bright star catalogue.It contains more that 9000 thousand stars of magnitude brighter than 6.5 with their HD number,RA,Dec etc. The B, V magnitudes for the stars have taken from the Hipparacus catalogue and FUV and NUV fluxes in micro Jansky along with their corresponding magnitudes in AB system have been taken from GALEX All Sky Survey.Hence the four observed magnitude and the colours are obtained for the selected bright stars.

[edit] Unit conversion from micro Jy to erg/cm2/sec/A

λFλ [Wm-2 μm-1]= νFν[Wm-2 Hz-1]

Fλ [Wm-2 μm-1]=c/λ2 [Hz/m]10-6 Fν[Wm-2 Hz-1]

1Jy = 10-26 Wm-2Hz-1

suppose Fν= X μJy and effective wavelength λeff = Y Angstrom then


Fλ = 3X/Y2 * 10-10 W m-2 μm-1

1 Wm-2 μm-1 = 0.1 erg cm-2 sec-1 A-1

Fλ= 3X/Y2 * 10-11 erg cm-2 sec-1 A-1

Where Y is in angstrom and it is the effective wavelength corresponding to FUV and NUV band of each spectral type and X is in micro Jansky.

[edit] Conversion of B & V magnitude into flux

According to the definition of magnitudes,we have:

m2-m1= -2.5*log(f1/f2)= -2.5*log(f1)+2.5*log(f2)

If m2 is the magnitude that corresponds to flux density f2 and in particular if m2=0 for that known flux,then 2.5*log(f2) becomes equivalent to an absolute zero point for the magnitude.The ultimate standard and reference for all broad band photometry is the star Vega whose apparent magnitude is zero.Now substituting the flux density in a given filter of Vega for f2 and putting m2=mvega=0,

log(f*) = -0.4*m*+log(fVega)

where m* and f* are the apparent magnitude and flux density of a star and

fvega for B filter (4400A) = 6.61*10-9 erg/cm2/sec/A

fvega for V filter (5500A) = 3.64*10-9 erg/cm2/sec/A

[edit] ANALYSIS

[edit] Observed Spectral type

We obtained the fluxes in all four bands viz FUV,NUV,B,V for our sample of stars.Then we determined the actual spectral types and extinctions of these stars.First, we took the observed V magnitude and an arbitrary spectral type and extinction to get the spectral energy distribution of a star. Then verified whether the measured fluxes of the photometric bands corresponds to their respective effective wavelengths in the spectrum of the star. Then we went on changing the spectral type and extinction till the fluxes fits to the spectrum corresponding to their effective wavelength with minimum chi square value.Here is the table:2 containing all the details of the unknown sources.

[edit] Expected spectral type from Colour-colour comparision

The observed colours of the stars are compared with the theoretical colour of the known stars.First the theoretical colours; B-V vs FUV-NUV of known stars of specific extinction=0.0 are plotted.The observed colours of unknown stars are over plotted on the theoretical colour-colour plane.Then the extinction is added to the theoretical colours in an interval of 0.1 till it coincides with the observed colours.In the sense that each point of the theoretical colour-colour plot corresponds to a spectral type of specific extinction and it gives the spectral type and extinction of an unknown star when it fits to a point of observed colour-colour plot of unknown stars.By this method of colour-colour comparision, the spectral type and extinction of all the observed stars are determined.Table:3

[edit] Verification

Next is to affirm the spectral types and extinctions of the observed stars, we determined from colour colour comparison . In order to substantiate it, we match spectrum of stars obtained by colour colour comparison with the actual spectrum obtained earlier and determined the chi square value for the matching.

[edit] FACTS ABOUT GALEX DATA - Brightness Limits

GALEX detector safety requirements limit observations of targets, that are brighter than the following values:

FUV: 5,000cps or mAB = 9.5 or Fl = 7 x 10-12 erg cm-2 s-1 Å-1 in the FUV;

NUV: 30,000cps or mAB = 8.9 or Fl = 6 x 10-12 erg cm-2 s-1 Å-1 in the NUV.

[edit] GALEX pointing centers must be separated from bright stars by :

0.75 deg for an object with Fl_NUV = 1 x 10-12, or mAB = 10.8 (5,000 cps)

0.88 deg for an object with Fl_NUV = 1 x 10-11, or mAB = 8.3 (50,000 cps)

1.00 deg for an object with Fl_NUV = 4 x 10-11, or mAB = 6.8 (200,000 cps)

1.50 deg for an object with Fl_NUV = 1 x 10-10, or mAB = 5.8 (500,000 cps)

2.00 deg for an object with Fl_NUV = 2 x 10-10, or mAB = 5.0 (1,000,000 cps)

(Fluxes and magnitudes in NUV band (~ 2300 Å), Fl in ergs cm-2 s-1 Å-1 )

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