Scoring rubric
Each problem is worth 30 points; presentation/reproducibility contributes 10 points across the set. Award credit for correct setup and units before the final number. Identify arithmetic slips separately from conceptual errors. Require correct comparative direction and an honest account of approximation gaps.
Problem 1: Radio velocity
Worked solution: v_radio=c Delta-nu/nu_0=105.5 km/s. A 10-kHz channel spans 2.111 km/s. This first-order convention is suitable for Galactic speeds and must be tied to a stated reference frame.
Show the governing equation, substitute with consistent units, retain intermediate values, and state which approximation limits the inference. Accept equivalent algebra and convention choices when explicitly stated.
Points: model and derivation 12; units and numerical work 10; uncertainty/limitation 4; interpretation 4.
Problem 2: H I column
Worked solution: The integral is 120 x 8=960 K km/s, giving N_HI=1.823e18 x 960=1.75e+21 cm^-2. For appreciable opacity, the thin estimate is a lower limit; absorption spectra can constrain optical depth.
Show the governing equation, substitute with consistent units, retain intermediate values, and state which approximation limits the inference. Accept equivalent algebra and convention choices when explicitly stated.
Points: model and derivation 12; units and numerical work 10; uncertainty/limitation 4; interpretation 4.
Problem 3: Line inference audit
Worked solution: The Gaussian integral is 1.0645 x 80 x 12=1022 K km/s; N_HI=1.863e+21 cm^-2. The rectangular estimate is 6.059% lower, not higher, than the Gaussian integral.
Show the governing equation, substitute with consistent units, retain intermediate values, and state which approximation limits the inference. Accept equivalent algebra and convention choices when explicitly stated.
Points: model and derivation 12; units and numerical work 10; uncertainty/limitation 4; interpretation 4.
Common errors
- Converting MHz/GHz or arcseconds/radians incorrectly.
- Confusing per-beam flux with integrated flux or brightness temperature with kinetic temperature.
- Treating flux-scale uncertainty as independent for every source.
- Using a linear Doppler approximation outside its regime or silently changing velocity convention.
- Calling a synthetic record an observation or reporting a completeness correction without its selection assumptions.