2. POLÍTICA GUBERNAMENTAL DE DERECHOS HUMANOS
2.1 POLÍTICA DE DERECHOS HUMANOS
2.1.3 Protección a sectores vulnerables
Now that we have discussed bound states and the equations of statistical equi- librium we will return to the transfer equation and describe how a self-consistent solution to this combined system of equations can be found. Beginning with the transfer equation as derived, we proceed to split the emissivity and the absorptivity into two components each
χν =χabsν +χ scat ν (2.71) ην =ηνemis+χ scat ν (2.72)
The absorptivity has two components, absorption and scattering, the emissivity is split into two terms, the emission and the scattering. Using these expressions for the source function
Sν = ηemis ν +χscatν χabs ν +χscatν (2.73) Here we will define a new concept, the photon destruction probability
ν = χabs ν χabs ν +χscatν (2.74) This is the probability that a photon which is absorbed not be re-emitted, and so
is converted to thermal energy. From this we get Sν =ν ηemis ν χabs ν + (1−ν) ηscat ν χscat ν (2.75) For isotropic scattering ηscat
ν /χscatν = Jν. If the emission is thermal then ηemis
ν /χabsν =Bν(T). Hence we can write the transfer equation as ∂Iν
∂τν
=νBν(T) + (1−)Jν −Iν (2.76)
Now we introduce the Lambda Operator1 (Cannon, 1973)
Jν = Λ[Sν] (2.77) Λ[Sν] = 1 2 Z ∞ 0 SνE1|t−τ|dt (2.78) where E1 is the first exponential integral. From this operator we construct the Lambda Iteration,
Sn+1
ν =νBν(T) + (1−ν)Λ[Sνn] (2.79)
This is the simplest form of the Lambda Iteration, where we successively apply the operation until we reach convergence of the source function. This iterative process of back-substitution of the previous iterate is referred to as a first-order, relaxation method, and this method will converge extremely slowly. Each iteration corresponds to the photons moving by one mean free path in the medium, and, in a medium of high optical depth and low photon destruction probability (high thermalization length) many iterations are required. This method can be vastly improved upon by what is known as the Accelerated Lambda Iteration (ALI). In order to see how this method can be employed we will recast the Lambda Iteration as a matrix operation, dropping the subscripts for notational convenience,
Sn+1 =B+ (1−)ΛSn (2.80)
where Λ is a matrix whose elements determine the coupling between each point (i.e. a function of optical depth). This equation is identical to the previous integral
formalism, and again will converge very slowly in the (standard) case that <<
1, where a photon may undergo a long chain of consecutive scatterings and a correspondingly large number of iterations are required. We can solve directly for
S
S= [1−(1−)Λ]−1B (2.81)
however performing this operation requires inverting the Λ matrix, which is im- practical.
At this point we introduce a method known as operator splitting. This method has a long mathematical history and was introduced into the field of radiative transfer in a very important paper by Olson et al. (1986), often referred to as OAB.
Λ= (Λ−Λ∗) +Λ∗ (2.82)
where Λ∗ is an approximate Lambda operator with an easily computable inverse. Hence we arrive at
Sn+1 = [1−(1−)Λ∗]−1[B+ (1
−)(Λ−Λ∗)Sn] (2.83) where, as we can see, the only inverse to be computed is the (computationally inexpensive) inverse of the approximate operator.
We can cast this equation in a somewhat different form by the introduction of a new source function, obtained from the old source function (known as the Formal Solution source function)
SFS =B+ (1−)ΛSn (2.84)
This definition provides a new way to write the iteration scheme, and hence a new insight into its structure,
δSn =Sn+1−Sn = [1−(1−)Λ∗
]−1[SFS−Sn] (2.85)
We can see from this equation that the factor which determines the iterative change is the difference between the old source function and the source function determined from the formal solution, and the iteration is accelerated by the [1− (1−)Λ∗]−1 factor. We note at this point that there is nothing significant about
applying this operation to the source function, a similar scheme could be a applied to the mean intensity.
Despite the fact that we are using an approximate operator it is clear from Eqn. 2.83 that a converged solution will be exact. If we set Λ∗ = 0 then we will recover the original Lambda Iteration, andΛ∗ =Λwill provide the exact solution, at the expense of the costly matrix inversion.
The selection of an approximate operator, Λ∗, is a difficult matter. Our re- quirements are that the operator have an easily computed inverse, and be an approximation of the full operator. There are a number of possible choices for the approximate operator, however OAB, in their foundational analysis, presented a set of strict mathematical criteria which determine the suitability of an approxi- mation (based on the convergence speed-up attained, rather than the physics of the problem). The most common choice of approximate operator is
Λ∗ = diag(Λ) (2.86)
This is often referred to as the local operator, as its application will provide the radiation field at a point as a result of the source function at that point, ignoring the contribution of the rest of the medium. Using this approximate operator we remove the self-coupling term from the iteration (we are only “iterating” photons on a scale appropriate to the medium). This makes the equations local and hence makes the solution far easier to determine (the equations remain non-linear, this is a fact of the physics of the problem). The diagonal operator can often be evaluated analytically, for instance using the two-stream approximation in a 1-D slab, in the case that ∆τ >>1, diag(Λ)≈1− 2 ∆τ2 (2.87) Sn+1 = + 2 ∆τ2 −1 [B+ (1−)(Λ−Λ∗)Sn] (2.88) which equates to a speed-up in convergence of 1/(+2/∆τ2) (Hamann, 1985). More complex approximate operators may be employed with a tridiagonal matrix being commonly used (Olson & Kunasz, 1987a). This operator introduces a degree of “non-localness” into the equations, by coupling each point to its neighbours. This
operator is useful for radiative equilibrium cases, as it natively supports radiation diffusion.
The iterative scheme outlined above is known as the Jacobi Method, and the generalised algorithmic process has the following form (Hamann, 1985),
• For a general linear system we begin with:
Ax=b (2.89)
• We break our matrix into three components, the diagonal D, the upper triangle−U, and the lower −L (note the sign convention)
A=−L+D−U (2.90)
• From this we obtain the iteration scheme (hereD is known as the precondi- tioner)
Dxn+1 =b+ (L+U)xn (2.91)
which can be rewritten as
δxn+1 =D−1(b+ (L+U)xn)
−xn (2.92)
• This process is repeated until δxn+1/xn is below some threshold value.
This method is very efficient as it requires no matrix inversions.
In order to improve the convergence of this method it is altered to employ a convergent Newton-Raphson scheme. The technique of Complete Linearisation, introduced by Auer & Mihalas (1969), is effectively a multi-dimensional Newton- Raphson scheme. This method has been superseded by the ALI, however this Newton-Raphson (or the Quasi-Newton) formalism remains important to these implementations, as is the idea of approximating (linearising) the equations in order to quickly generate an inaccurate solution, followed by successive iteration to reduce the error.
In Complete Linearisation, the Newton-Raphson method enters the iteration through the rate equations (Peraiah, 2001). The iteration proceeds as follows: the
Lambda Operator provides the mean radiation field from the source function, and from the mean radiation field, through the rate equations, we can compute the populations. From these populations a new set of populations can be determined. Hence, very loosely, we have
Jk = Λ[Sk] (2.93) nk =Pij(Jk)−1b (2.94) nk+1 =nk − Pij(J k) ∂Pij(Jk)/∂nk (2.95) The application of the Newton-Raphson method provides the the subsequent iterate of the populations from which we can construct a new source function. This loop is repeated until the populations converge. However in multi-level ALI implementations the repeated inversion of the Jacobian required to perform this calculation may be costly, as well as being difficult to implement, and there are a number of techniques which avoid this.
Another important development is the careful preconditioning of the rate equa- tions to ensure numerical accuracy and linearity. Photons in the wings of a spectral line will encounter a lower optical depth, and will travel a greater distance between extinctions than line core photons. As such these wing photons play a greater role in the non-locality of the transfer problem, and removing the core photons im- proves the numerical stability of the solution schemes at little cost. Hence we precondition the rate equations, employing the “core-saturation” method outlined in Rybicki (1972); Rybicki & Hummer (1991), which, as well as ensuring linear- ity, removes the effect of passive scatterings from the high optical depth line-core. This process of preconditioning is essentially equivalent to linearisation. This was demonstrated by Socas-Navarro & Trujillo Bueno (1997), who showed that pre- conditioning effectively takes into account the linear response of the radiation field to perturbations in the source function (Hubeny & Lanz, 2003).
In this thesis we will make use of two implementations, MULTI (Scharmer & Carlsson, 1985), and RH (Uitenbroek, 2001), both of which are based on the ideas we have discussed to this point. In the case of MULTI the iterations begin with an estimate for the populations, nk and the radiation field Jk which determines
Pij, X j6=i nk jP k ji−n k i X j6=i Pk ij =−E k (2.96)
whereEkis the error at this iteration. We require a new set populations and rates,
of the form nk+1 =nk+δnk (2.97) Pk+1 ij =P k ij +δPijk (2.98)
such that the error term goes to zero. Substituting this equation into the sta- tistical equilibrium equation (linearising the equation by neglecting all cross-term perturbations) we get X j6=i nk jδP k ji+δn k i X j6=i Pk ij − X j6=i δnk iP k ij − X j6=i nk iδP k ij =−E k (2.99)
In order to determine the appropriate change to the populations we must de- termineδPk
ij in terms ofδnki. SinceδCij = 0 (the rate equations only couple to the
radiation through the radiative rates) we determine the perturbation in the rate equation from the perturbation in the mean radiation field
δPk
ij =BijδJk (2.100)
To complete this set of equationsδJk can be determined from the populations.
In MULTI this calculation is carried out by the Scharmer Operator, which is a form of the Lambda Operator which returns the outgoing intensity,
I+ ν = Λ † ν[Sν] =eτν Z ∞ τν Sν(tν)e−tνdtν (2.101)
This equation is then solved using single point quadrature (Scharmer, 1981). Since these equations will not result exactly in En going to zero, due to the
linearisation, iteration is required.
In the case of RH, based on the method outlined in Rybicki & Hummer (1991), the equations are preconditioned, resulting in a linear expression for the popula- tions. RH also makes use of a different operator, the Ψ operator, which is defined
as follows
Jν = Ψν[ην] (2.102)
which is to say the Ψ operator differs from the Λ operator by a factor of 1/χ. This operator has the benefit of being linear in the populations, where the Λ operator is not. Notwithstanding these differences in the operator and the preconditioning, the RH implementation (Uitenbroek, 2001) reaches a solution by effectively the same method as was outlined in the case of MULTI.