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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. General ERT:. s = path along a ray and. Spherical Geometry: Steady state, polar coordinates d r = cos q d s , r d q = -sin q d s :.

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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 General ERT: s = path along a ray and Spherical Geometry: Steady state, polar coordinates dr = cos q ds , rdq = -sin q ds :

  3. Using Sn = jn/an: VERY difficult to solve. Assume atmospheres thin compared to stellar radius Validity: Sun: h ~ 150km, R ~ 105 km Plane Parallel Geometry: Use for rest of “traditional” RT theory. dq/dr = 0, radial optical depth, dtn = -knrdr : Solutions…

  4. ERT: Moment Equations Approximations: analytic/semi-analytic solutions. Assume Sn isotropic: form moment equations. Angle averaging and apply

  5. Multiply by m before averaging: 0 for isotropic Sn Substitute into previous moment equation:

  6. q t1 (t0 - t1)/m = Optical Depth q t0 Formal Solution of ERT Sn isotropic Multiply by integrating factor, exp(-t/m) (n-dependence implied):

  7. Two regimes: Outward (m > 0), Inward (m < 0). Boundary conditions: t0g infinity No inward illumination, In(m < 0) = 0 In measures Sn weighted by exp(-t/m) along beam up to point of interest

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

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

  10. In similar vein: Surface Values The emergent intensity and flux at the stellar surface are:

  11. 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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