Problem 5.1: worked solution

Solve dI/dtau=S-I by multiplying by e^tau: d(Ie^tau)/d tau=S e^tau. Integrating from 0 to tau gives I(tau)=I0 e^-tau+S(1-e^-tau). For I0=0.2, S=0.7, tau=1, I_exact=0.51606028; DOP853 gives 0.51606028, absolute solver difference 9.48e-14. This checks the numerical integration against the analytic solution, not the constant-source slab against an atmosphere. An integrating-factor derivation or direct substitution into the ODE is acceptable; include the boundary condition.

Problem 5.2: worked solution

Using I0=0.2 and S=0.7, tau=0.1: exact=0.24758129, thin=0.25000000, absolute error=0.00241871, relative error=0.9769%; tau=5: exact=0.69663103, thin=2.70000000, absolute error=2.00336897, relative error=287.5796%. The computed first crossing of 1% relative intensity error is tau=0.101317. Check the root by evaluating the error immediately below and above it. Because I_exact is the relative-error denominator, this threshold is specific to these boundary/source values. A Taylor-remainder bound is an acceptable cross-check, but the numerical criterion must still be evaluated on the stated observable.

Problem 5.3: worked solution

In a non-LTE atmosphere the source function is not generally the Planck function B_nu(T); it depends on level populations and the radiation field. For isotropic coherent-scattering illustration, S_nu=(1-epsilon)J_nu+epsilon B_nu, where epsilon is the photon-destruction probability and J_nu is the angle-averaged intensity. Non-grey opacity makes optical depth frequency-dependent; anisotropic scattering, redistribution, geometry, and coupled statistical equilibrium require an angular/frequency closure and boundary conditions. A constant grey S cannot represent these feedbacks. Accept a different scattering source-function derivation if it identifies the coupling and the approximation being relaxed.

Scoring guide

Each problem is scored for model/setup (25%), correct computation or derivation (30%), validation and uncertainty (25%), and physical interpretation/communication (20%). Accept alternative correct methods when they meet the same evidence standard. See assessment instructions for error classification and resubmission guidance.