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6. Formal Solution of ERT

6. Formal Solution of ERT. Rutten: 4.1 Formal solution of plane-parallel ERT Exponential integrals and operators. Formal Solution of ERT. The general equation of radiation transfer is:. where s is the geometrical path along a ray and.

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6. Formal Solution of ERT

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  1. 6. Formal Solution of ERT • Rutten: 4.1 • Formal solution of plane-parallel ERT • Exponential integrals and operators

  2. Formal Solution of ERT The general equation of radiation transfer is: where s is the geometrical path along a ray and Spherical Geometry: Assuming steady state (time independence), adopting polar coordinates with dr = cos q ds and rdq = -sin q ds and taking a spherical star with azimuthially symmetric intensity we get:

  3. and with Sn = jn/an we get: This equation must be solved for stars with extended atmospheres. It is very difficult to solve and only recently have stellar atmosphere calculations been performed for this geometry. Most often it is assumed that stellar atmospheres are thin compared to their radius, so that the plane parallel approximation holds. Plane Parallel Geometry: This is the approximation that we will use for the remainder of the discussion of “traditional” radiation transfer theory. In this approximation dq/dr = 0 and, with the radial optical depth, dtn = -knrdr, we get We will now spend some time on finding solutions to this equation…

  4. But first, how valid is the plane parallel approximation? For Sun, scaleheight, h ~ 150 km, radius R ~ 105 km

  5. ERT: Moment Equations Due to the complexity of the ERT, approximations are made to allow us to simplify the equations and find semi-analytic solutions. If we assume the source function, Sn, is isotropic we may form various moment equations. First, perform angle averaging and apply

  6. If we multiply both sides by m before applying the angle averaging we get: 0 for isotropic Sn If we substitute this into the previous moment equation we get

  7. Formal Solution of ERT Sn isotropic Multiply by integrating factor, exp(-t/m) (n-dependence implied):

  8. q t1 (t0 - t1)/m = Optical Depth q t0 Split into two regimes: Outward (m > 0), Inward (m < 0). Boundary conditions: t0g infty. No inward illumination, In(m < 0) = 0 Intensity measures the source function weighted by exp(-t/m) along the beam up to the point of interest.

  9. Moment Equations, Exponential Integrals, Operators The exponential integrals En are defined by (Rutten 4.1.2) Tabulated in textbooks. For this course, we’ll need approximations at small t, so use

  10. Schwarzschild-Milne Equations The Schwarzschild equation for the mean intensity: The Milne equation for the flux:

  11. Completing the intensity moments in terms of exponential integrals, we get for the Kn integral: Surface Values The emergent intensity and flux at the stellar surface are:

  12. Operators We can write the above equations in terms of operators. For the specific intensity we use the Laplace Transform: In stellar atmospheres theory an important operator is the classical Lambda Operator, Lt, defined by the RHS of the Schwarzschild eqn: The F and c operators are:

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