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4.1 OPACITY

4.1 OPACITY.

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4.1 OPACITY

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  1. 4.1 OPACITY As photons pass through matter they can be scattered or absorbed. This coupling between the radiation and matter causes a modification of the radiation field. The probability of absorption depends on the frequency of the photon. For a photon passing through a slab of material thickness, dx, the probability that the photon is scattered or absorbed is kn, the volume opacity, which has units of area per unit volume ( units m-1 ). The specific opacity is defined as kn and has units of m2 kg-1 We may also define the following ; a) The mean free path, i.e. the average distance a photon travels between interactions as ln = kn-1 b) The opacity per scattering particle (cross section) as sn = knr/n where n is the number density of particles per unit volume. c) The relationships between the opacities are kn = knr = nsn d) The total loss of intensity from the beam which traverses a thickness dx (m) is dIn = - kn r Indx NOTE that the units kn rdx must become dimensionless e) The optical thickness (ratio of the distance travelled to the photon’s mean free path is When tn >> 1it is an optically thick material and when tn << 1it is an optically thin material PHYS3010 - STELLAR EVOLUTION

  2. The opacity involves all the microscopic processes which contribute to the absorption of the radiation. the major processes are : Bound-Bound Absorption. Only photons of fixed energies which correspond to the electronic transitions are absorbed. This is not important in an ionised region. Bound-Free Absorption. The electron is ejected from a bound state to a free state. The importance obviously depends upon the number of bound electrons. Free-Free Absorption. The electron is moved from one free state to another. The cross-section is larger for lower energy photons. Scattering. The photons scatter from electrons by the Compton process. This is not a true absorption of photons, but rather slows the rate of energy escape down. NUMERICAL VALUES OF OPACITY - KRAMER’S LAW The calculation of opacity is extremely complicated. All the photon energies, atomic types, densities, states of ionisation etc. etc. have to be taken into account. All of these vary, not only from star to star but throughout the envelope. Modern methods use sophisticated Monte-Carlo computer programmes to study the photon transport. Such methods are not so useful for teaching processes. In this course mostly the historical analogue expressions are used since they offer a better insight into the underlying physical processes, and their relative importance. NOTE Log r Log kn • The opacity is low at both low and high temperatures • At high temperatures the photons have high energies and are not easily absorbed • At low temperatures most of the atoms are not ionised and there are few free electrons to scatter the photons 2 1 Log T 7 4 6 5 KRAMER’S EMPIRICAL LAWS • At T > 105 K, the expression k = k0 rT-3.5……………..(1) • provides a very useful approximation for order of magnitude calculations. • At T ~ 104 K, we may use k = k0 r1/2 T4…………....(2) PHYS3010 - STELLAR EVOLUTION

  3. Log T 15 M¤ ~ 105 y 4 M¤ ~ 106 y 2 M¤ ~ 107 y 4.2 T-TAURI STARS Observationally T-Tauri stars are cool stars which lie above the main sequence. They are associated with the dense interstellar star forming clouds. They exhibit rapid irregular variability and have violent convection taking place in their envelopes. They are thought to be stars approaching the main sequence H -> He burning phase. Log L In the initial phases dissociation and ionisation of the gas makes g< 4/3 and collapse follows which is governed by tff. The luminosity falls rapidly as the size decreases with little change in temperature (~isothermal). Later when the opacity increases the energy is stored and the temperature and luminosity increase. When a radiative core develops the star moves onto the main sequence. Typical timescales are Hayashi Tracks 4.3 HERBIG-HARO OBJECTS Hubble picture of jet from a protostar which is hidden in a dust cloud near the left edge of the image. Hubble picture of protostellar object which reveals an edge-on disk of dust encircling the newly forming star. Observations of star forming regions have shown that the IR sources are generally associated with bipolar outflows which are well collimated jets directed in opposite directions from the central source, assumed to be a very young protostar. (See Herbig -Haro objects e.g. ARAA 1985, 23, 307) PHYS3010 - STELLAR EVOLUTION

  4. 4.4 A PICTURE OF STAR FORMATION. The figure below (taken from ARAA, 1987, 25, p.72) represents an attempt to incorporate all the available information into one general picture. a) Condensations form within the dust/gas clouds due to gravitational collapse. b) A protostar with an accreting core is formed. The angular momentum inherent in the infalling matter causes a rotating disc to form with its plane perpendicular to the rotation axis. The gravitational energy of the infalling material is reradiated in the IR which escapes since the protostar is transparent at this stage. c) Gradually the infalling material will fall preferentially onto the disc and the stellar wind rushes out along the rotation axes of the system, creating the bipolar flows and collimated jets. d) Eventually, when all the available matter is accreted, the system is left with a young newly formed star having a circumstellar disc. All of the above phases are supported by imaging and/or spectral observations. However it is also possible to imagine that the nebular disc will disappear as matter becomes incorporated into planets or stellar companions. The above picture is representative of stars at the lower end of the mass scale. More massive stars will start hydrogen burning and hence join the main sequence whilst accretion is still taking place. PHYS3010 - STELLAR EVOLUTION

  5. dm dr Density r(r) R r p + dp m(r) p p + dp p dm 5. STELLAR STRUCTURE 5.1 HYDROSTATIC EQUILIBRIUM On the main sequence a star is a hot ball of gas, it stays there a long time with a negligible amount of change taking place. Why is it so stable? There is a self-adjusting and precise balance of two physical forces : Gravitational attraction and gas pressure. Consider a unit area element dm within the star at a distance r from the centre, there will be a pressure difference across this element of dp. The mass of the element is dm = rdr The force of attraction between dm and the rest of the material [mass m(r)] within the radius r is In equilibrium Thus MASS RELATIONSHIP Now the mass within radius r is Giving the differential equation PHYS3010 - STELLAR EVOLUTION

  6. s is the cross-section { n is the number of atoms/m3 Where kis the opacity ENERGY TRANSPORT THROUGHOUT THE STAR Stars are bright sources of electromagnetic energy. The energy is generated within the star, and as we we will see later, predominately towards the centre of the star. This energy, which is initially in the form of X- and g-rays, must travel from the innermost regions to the outer surface where it shines away to deep space. How does it get there? - Let us look at some possibilities. 5.2 RADIATIVE TRANSPORT - EQUATION OF RADIATIVE TRANSFER Deep within the star the temperature is extremely high. The radiation density is also high but, because of the high opacity of the overlying layers, the net outward flux will be relatively small. In this case, because of the vast number of scatterings which take place for the outward moving photons, it is possible to use a diffusive approximation. At equilibrium the radiative energy flux is Where D is the diffusion constant and Ur the radiation density Where l is the photon mean free path Now As discussed above For the case of black-body radiation PHYS3010 - STELLAR EVOLUTION

  7. { Where a is the Black-Body constant The corresponding radiation density is s is the Stefan-Boltzmann constant Giving Thus is the equation of radiative transfer and can be written as a differential equation : NOTE i) The opacity of the stellar material determines the radiation flux through the star and hence the luminosity of the star. High opacity means low luminosity. ii) The radiative flux depends on the temperature gradient. Zero gradient means zero flux. ROSSELAND MEAN OPACITY It is possible to define an average opacity which is independent of frequency PHYS3010 - STELLAR EVOLUTION

  8. In dW n dA 5.3 RADIATIVE EQUILIBRIUM The force exerted on a small volume element by means of radiation pressure across a spherical shell is and is analogous to the hydrostatic equilibrium condition. The pressure is the rate at which momentum is transported across unit area. The momentum of a photon of energy E is E/c, so that a beam of intensity In (J m-2 s-1 Hz-1 sr-1), carries a momentum flux of In/c in the direction of the beam The component of the momentum flux perpendicular to an area dA is In cos q /c. Thus the pressure across the area in a direction perpendicular to the plane is The associated radiation pressure is Giving And since Thus Using the equation of radiative transfer we obtain Which describes RADIATIVE EQUILIBRIUM PHYS3010 - STELLAR EVOLUTION

  9. LUMINOSITY RELATIONSHIP If, mainly by thermonuclear processes, an amount of energye(r) (J kg-1 s-1)is being generated per unit mass of the star at a distance r from the centre, then in traversing a shell of thickness dr the additional amount of energy added to the radiation field is So that This is usually called the energy generation equation The generation of energy e(r) is strongly dependent on the local physical conditions, mainly temperature, density and composition. STABILITY IN RADIATIVE EQUILIBRIUM We have seen that under the conditions of radiative equilibrium the luminosity of the star is not determined by the rate of energy generation by e.g. nuclear processes but rather by the radiative transfer condition. Thus if the total nuclear generation is less than the loss by radiation this net loss will be made up by contraction of the star. One half of the gravitational potential energy goes into radiation and the rest goes into heating the stellar material. Since (see later) thermonuclear reactions are highly temperature dependent then this heating will rapidly restore the rate of nuclear burning. Should the rate be too high then the star will expand and cool adiabatically. Hence there are inbuilt regulatory mechanisms which ensure stability for which PHYS3010 - STELLAR EVOLUTION

  10. 5.4 EDDINGTON LUMINOSITY LIMIT When radiation is absorbed, an element of matter has momentum imparted to it and will be accelerated. So that if we consider an element of unit area situated at a distance r from the centre of the star then the outward force acting on this element due to the radiation pressure will be The inward gravitational force acting upon the same element is If the radiation pressure is high we may envisage the situation in which it is comparable to the gravitational force exerted on the outer layers of the star. i.e. Giving This is called the Eddington Luminosity If L > LEdd the outer layers of the star will not be in hydrostatic equilibrium. Instead they will accelerate outwards under the radiation pressure and mass loss from the star will result. (Note that k has units of m2/kg) PHYS3010 - STELLAR EVOLUTION

  11. 5.5 CONVECTIVE ENERGY TRANSPORT If the opacity within the star is very high, radiative equilibrium becomes impossible and heat will be transported through the star by convection. Consider a cross-section in the star in which a small gas bubble is displaced a distance dr without any internal energy losses. * * P2 r2 P2r2 Before the displacement dr After perturbation the pressure is at equilibrium i.e. P1 r1 However the density within the element may differ from that of the surrounding material For adiabatic motion * * P1 r1 So that For a perturbation to be unstable and move upwards Substituting in order to remove * terms PHYS3010 - STELLAR EVOLUTION

  12. Assume a small perturbation Expanding Where k is the Boltzmann constant For an ideal gas Differentiating At equilibrium substituting We get Giving the equation of CONVECTIVE EQUILIBRIUM The criterion for convection is PHYS3010 - STELLAR EVOLUTION

  13. 5.6 LUMINOSITY LIMIT FOR CONVECTION Convection will occur if the actual temperature gradient exceeds the adiabatic temperature gradient i.e. Now and We also have for an ideal gas Wheremis the mean molecular weight And for hydrostatic equilibrium At the limit we obtain Giving NOTE This is the maximum luminosity possible at radius r without convection. i) Convection is more likely when a) The opacitykis high b) The luminosity is high c) g/(g-1) is large ii) This often occurs close to the centres of stars where m(r) is small and ionisation causes the opacity to be large PHYS3010 - STELLAR EVOLUTION

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