ASTR 474 Lab 05: Radiative Transfer and Approximation Domain

Purpose and learning objectives

L07-L08 · optical depth and formal solution. This investigation develops a numerical method and tests its output against a declared benchmark, invariant, uncertainty model, or independent implementation.

Run a specified scientific workflow from the pinned environment.
Measure an explicit numerical or statistical diagnostic with units and provenance.
Separate numerical error from physical model discrepancy and interpret a limitation.

Preparation, setup, and inputs

Software: CPython 3.14.6; dependencies in requirements.txt. From repository root run .venv/Scripts/python.exe materials/ASTR474/src/astr474_computations.py. Then use the corresponding CSV/JSON under materials/ASTR474/data/generated/. Required data: generated/computed_results.json.

Apparatus: computer with local CPU, terminal, text editor, and browser; no telescope, GPU, network account, or paid service is required. Preserve the generated input files and record file hash/version.

Model scope: The uniform slab is a synthetic controlled transfer benchmark; no atmosphere or observed spectrum is claimed.

Procedure

Implement dI/dtau=S-I with I(0)=0.2 and S=0.7 using SciPy DOP853.
Evaluate tau=0.01, 0.1, 1, and 5; compare each numerical result with the analytic constant-source solution.
Compute the optically thin first-order approximation from the same boundary data and plot relative error versus tau.
Vary source function and incoming intensity; identify any condition where a relative-error denominator makes the metric misleading.
Submit a short validity statement with a declared tolerance and optical-depth range.

Measurement record

Quantity	Value or field	Unit/convention	Source/method
Input configuration	Record solver, version, seed, timestep/grid/tolerance	As defined by the experiment	Command line and source file
Primary diagnostic	Record every resolution/sample/fit row, not just the preferred result	State units and normalization	Generated CSV/JSON
Independent reference	For the first-order thin approximation, relative intensity error is 0.000122 at tau=0.01 and 2.88 at tau=5; the tolerance defines whether either is acceptable.	As stated in the generated example	Analytic/JPL/invariant cross-check
Uncertainty/sensitivity	Repeat with changed resolution, seed, parameter, or method	Report interval or relative change	Student analysis

Analysis and uncertainty

Perturb tau and source function over stated ranges; distinguish numerical solver tolerance from approximation error.

Plot the diagnostic against the controlled parameter; label axes, units, normalization, and reference. Explain any non-monotonic point rather than smoothing or dropping it. If propagating independent errors, state why independence is plausible; preserve correlated/common-mode terms separately.

Deliverables and assessment

Runnable source or notebook export with command, package versions, and all input/output paths.
One scientific figure plus a machine-readable table of measurements and convergence/sampling/residual diagnostics.
A 700-1000 word technical report describing model, method, validation, uncertainty categories, and one failed assumption.
A short claim-evidence table separating result, numerical check, and domain limitation.

Rubric: 30% correct method and setup; 25% validation and numerical diagnostics; 20% uncertainty/provenance; 15% physical interpretation; 10% clarity and reproducibility. Revisions and resubmissions are encouraged under the syllabus mastery policy.

Data and source provenance

The uniform slab is a synthetic controlled transfer benchmark; no atmosphere or observed spectrum is claimed. Provenance levels, request records, and source verification limits are documented in reference-log.md; generated files and seeds are documented in data/README.md.

References: