Interstellar bubbles

From Tauwiki

1 Introduction to ISM and IS Bubbles
2 Our objective:
3 Plan of analysis
4 DATA Analysis
5 Bubble Nebula
- 5.1 For comments
6 References

[edit] Introduction to ISM and IS Bubbles

[edit] Interstellar medium

The space between stars contains significant amount of matter termed as "Interstellar matter", comprising of gas (atom, molecules,ions and electrons), dust, magnetic field and cosmic rays. This interstellar matter (or ISM) was first seen as dark zones in the first long exposure photographs of the Milky Way taken by Edward Barnard. Although not first recognized as such, these were later realized to be interstellar dust. Further evidence for interstellar dust came from extinction observations by Trumpler (1930) and, convincingly, from the polarization of starlight which, incidentally, also proved the existence of interstellar magnetic fields. The part of the interstellar medium, interstellar gas, was first discovered as "stationary absorption lines" by Hartmann (19xx) in the spectrum of spectroscopic binary δ Orionis.

Interstellar matter accounts for 10-15% of the total mass of the galactic disk. The distribution of the ISM is inhomogeneous at small and large scales with a greater concentration near the galactic plane and along the spiral arms. Roughly half the interstellar mass is confined to discrete clouds occupying only 1-2 % of the interstellar volume.

Edited up to here.

Gas in the ISM is found in one of the Five Thermal Phases.

Chemical composition of interstellar matter is close to the "cosmic composition" i.e., 90.8 %( by number) of hydrogen, 9.1% of helium and 0.12% of heavier elements. The study of interstellar absorption lines reveals that significant fraction of these heavier elements is often missing or "depleted" from the gaseous phase of the ISM. These depletion factors have been observed to vary across the sky. It is observed that depletion tend to be more severe in regions with higher density and lower temperature (Jenkins,1987; Van Steenberg and Shull,1988) they also seem to depend weakly on the degree of ionization , depletion is less severe in the warm ionized medium than in warm neutral medium (Howk and Savage, 1999) On average, the most common heavy elements like C,N,O are only depleted by factors of 1.2 - 3 , whereas refractory elements like Mg, Si and Fe are depleted by factors of 10-100 ( Savage and Sembach,1996). And also it is estimated that 0.5-1% (by mass) of interstellar matter is in the form of dust.

[edit] Interstellar bubbles

Interstellar bubbles are the structures formed by the interaction between the fast winds of stars and interstellar medium. These structure creates an inner cavity surrounded by thin shell of swept up ionized gas which in turn may surrounded by neutral atoms /molecules. Discoveries of high velocity gas (up to 70 t0 100 km/s) in several diffuse nebulae, in particular around Wolf Rayet (WR) stars, for the first time formally established idea of presence of thin nebular shell formed of two layers: the shocked stellar wind and the shocked ambient medium separated by a contact discontinuity.

Dyson & de Vries (1972) and Avedisova (1972) obtained similarity solution to this flow pattern and further developed by Falle(1975). Weaver and others (1977) made a comprehensive study of the flow pattern and gave a model which described the physical conditions and structure of bubble. The self similar solution gave the prediction which can be observed and hence it can be used to test the model.

Most of the early models of bubbles assume that the stellar wind interacts with the ambient medium through momentum transfer and find that the shell expansion follows $r\propto t^\frac {1} {2}$ , where r is the shell radius and t is the dynamic age.

Dyson and de Vries (1972) were the first to suggest that the expansion of a bubble is driven by the thermal pressure of the shocked wind and follows the $r\propto t^ \frac {3} {5}$ .

Castor et al (1975), considering the thermal conduction through the discontinuity region and gave the basic structure of an interstellar bubble, which is divided into four regions:

(1) Interior Region (a) consists of freely expanding stellar wind moving with a velocity ( $\approx$ 2000 km/s) enters shock which converts part of the energy into thermal energy

(2) The next region (b) contains shocked stellar wind where the sound speed is very high. It is the expansion of this hot gas, drives the shell of shocked interstellar gas and looses energy by doing this work. Velocity very quickly drops to a value less than the internal sound speed. The sound crossing time becomes much less than the expansion timescale. The total energy of the region is determined by the energy lost in doing work against its surrounding and the energy fed through the internal shock. Because the expansion velocity is much less than the sounds speed, the energy in this region is much less than the sound speed, the energy in this region is almost entirely thermal. Since radiative cooling is extremely low in this region, this region is spatially extended. The high temperature (≈ $106$ K) ensures that the elements are very highly ionized, which are not effective coolants and the cooling rate is appreciably lower.

(3)This region (c) consists of Swept up interstellar medium. Radiative cooling is extremely effective in this region; therefore immediate post shock temperature is considerably lower than behind the inner shock. The gas in this region contains ions which can be collisionly excited by electrons to metastable states hence giving forbidden lines; therefore cooling is very effective in this region. This cooling increases the compression behind the outer shock; hence this region must be extremely thin. The gas cools down to a temperature (≈ $104$ K) at which the cooling rate is equal to energy input rate due to photo ionization by the stellar radiation field. Therefore it has properties of a typical radiatively excited HII region. Since the region is thin, and low sound velocity it is essentially at constant pressure. Across this surface the temperature, density and other physical characteristics of the gas change discontinuously. It is usually called a contact discontinuity.

(4) last is the region (d) of surrounding ambient interstellar medium.

The temperature and density structure of an interstellar bubble is illustrated in figure below.

[edit] UV absorption line formation

Radiative Transfer in Lines

The transfer equation in case of UV (where hν>>kT) is given by

Where - $κ ν I ν$ is the absorption term.

Which gives

where Optical depth $τ ν$ is defined as

Where $n l$ is the density of absorbing atoms (molecules) in the line of sight at l. $f l$ is the “oscillator strength” which is defined as probability for an induced transition from the lower to the upper level.

where

Where H(a,u) is Voigt function given by

Using these equation we can find Column density

Equivalent width W is given by

[edit] Our objective:

In this analysis we want to understand the physical condition of the bubble and also study the evolution of these bubbles.hence to get information about their structure and try to understand how these objects evolve or form we observe the radiation coming from the central star of these bubbles and there by we get know about the absorption lines formed inside the spectra by which we can determine the physical condition of the bubble.

[edit] Plan of analysis

We observe the hot central stars known (/ believed) to be responsible for such kind of a structure and analyze their absorption spectra, hence understand the physical conditions, various atoms/ions present at the place of absorption. where data of other stars in the vicinity is available we will supplement the result of the analysis of those stars while intrepreting the spectra formed inside the bubble.

[edit] DATA Analysis

Data has been taken from IUE archive from Multimission Archival at STsci (MAST)

[edit] IUE Mission:

The IUE satellite was launched on 26 January 1978 into an elliptical geosynchronous orbit. The IUE scientific instrument consists of a telescope, acquisition camera, two ultraviolet spectrographs, and four cameras. The telescope is a 45-cm aperture, f/15 telescope of Ritchey-Chretien design. The long-wavelength spectrograph operates in a wavelength range of 1850 to 3300 Å. The short-wavelength spectrograph operates in a wavelength range of 1150 to 2000 Å. Each spectrograph has two dispersion modes. High resolution employs an echelle grating and cross-disperser, giving roughly 0.2 Å resolution. Low resolution employs the cross-disperser grating alone, and yields approximately 6 Å resolution. There are four cameras, two for each spectrograph. One prime and the other is Redundant. In the long-wavelength range, both the prime (LWP) and redundant (LWR) cameras were used during the mission. With the short-wavelength spectrograph, only the prime camera (SWP) was fully functional.

NEWSIPS (NEW Spectral Image Processing System) uses new algorithms and re-derived calibrations to process data taken by IUE. All images are processed with the same routines to create a `Final Archive’ of IUE data, making direct comparisons easier for observations of the same target. Among the many additional benefits of the Final Archive are a reduction in the noise level present in many spectra, an error estimate for each extracted point in the spectrum, and more accurate relative and absolute calibrations.

[edit] Code

we have used Interactive Data Language(IDL) for our analysis, we have written a custom program to analyze the data, we have choosen Voigt profile as the model profile and we have defined this model profile having four free parameters namely height, central wavelength, width and continuum. we obtained the Equivalent width by integrating the area under profile.

whereever more no of observations are available, we have added those spectra , together to increase the S/N ratio by taking the weights as square root of the exposure time.required confidence intervals are found for the different parameters using the method explained in Lampton's paper.

Part of the code is as below

dw=r(2) ; Doppler width parameter

a=(r(5)*r(1)^2)/(4*(!pi)*dw*c) ; r(1) is observed wavelength, r(5) is gamma value

u=((r(1)-x)/dw)

v=voigt(a,u) ; Viogt fn provided by idl

gauss,g,k ; instrument Gaussian model profile, to compensate for instrument broadening.

f=convol(v,k,total(k),/edge_truncate,/nan); convolution of Gaussian and Voigt

f=r(3)*exp(-r(4)*s) ; final model profile used to fit the data

where r(4) is height parameter of the profile,r(3) is continuum parameter.

[edit] Calculations

I have calculate the shift velocity by knowing central velocity for this I have taken the lab wavelength and f values from D.C.Morton’s paper (ApJ Supplement 77:119-202, 1991 September). We can find the temperature by knowing Doppler width ( $Δλ D$ ) using

And we can use the column density factor to calculate column density by using

Column density is given by

[edit] Equivalent width:

The Equivalent width of an absorption line is width that a line would have if it had a rectangular profile with zero intensity at the line centre, having area equal to the area of the profile.

For practical purpose it is as good as measuring the area of the absorption line profile under the local continuum ( $I λ,0$ ).

We can also find the column density by CURVE OF GROWTH method

[edit] CURVE OF GROWTH

The curve of growth is a graph showing how the equivalent width of an absorption line increases with the number of atoms producing the line.

In an optically thin gas (where all the atoms, even that are in back are easily seen), the equivalent width of an absorption line is linearly proportional to the number of atoms in the initial level of the line. This part of the curve is called as “linear part” of the curve.

When no of atoms goes on increasing equivalent width cannot continue to increase indefinitely and linearly, when intensity at the centre of the line becomes zero,increasing the no of atoms (column density) will make no difference at all to the central intensity,and hardly any change in equivalent width. Hence this part of the graph will become almost flat and hence known as “logarithmic part” as

When yet further atoms have been added, though the central depth cannot get more deeper, the wings of the profile start to add to the equivalent width , so that equivalent width starts to increase again rather slowly than in optical thin case. Thus this part of the curve is called as “square root part“since here,

[edit] How I constructed theoretical curve of growth

First I defined model profile (Voigt function) which has column density (N) as parameter, and I found equivalent width ( $W λ$ ) of the model profile for various N values. To scale the plot for iron lines I divided $W λ$ by lab wavelength(λ) of the weak Fe II and multiplied f value and lab wavelength(λ) of the same line with N, and plotted log(W/ λ) along y-axis and log(Nf λ) along x-axis.

Part of the code written to obtain theoretical curve of growth

a=(1e-4*1^2)/(4*(!pi)*dw*c)

u=((1-x)/x)*(1/dw)

v=voigt(a,u)

y=[1e-7,1e-6,1e-5,1e-4,1e-3,1e-2,1e-1,1e0,1e1,1e2,1e3,1e4,1e5,1e6,1e7,1e8,1e9,1e10,1e11,1e12,1e13,1e14,1e15]

n=n_elements(y)

z=findgen(n)

for i=0,n-1 do begin

f=1*exp(-y(i)*v)

z[i]=int_tabulated(x,1-f)

endfor

print,z

plot,alog10(y*1E-10)-10.0074,alog10(z),background=255,color=0

[edit] Method to obtain column density (N) from equivalent width

We can obtain column density (N) from the observed equivalent width by plotting theoretical curve of growth taking log(Nfλ) along x-axis and log(W/λ) along y-axis and sliding horizontally along the empirical curve of growth for observed values while plotting Log(fλ) on x-axis.

[edit] Cons of curve of growth method

In general interstellar absorption lines may contain a mixture of both saturated and unsaturated components, in the case where lines are heavily saturated or show measurable damping wings, the equivalent width curve of growth method is unreliable because one cannot tell where the continuum should be placed, which leads to large systematic errors in measuring $W λ$ .