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: Noise scaling
Worked solution: There are n_pol Delta-nu tau independent samples, so their rms falls as the inverse square root: sigma_S=SEFD/sqrt(n_pol Delta-nu tau). Substitution gives sigma_S = 3.536 mJy.
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: Brightness temperature
Worked solution: Omega_b=pi theta^2/(4 ln 2) and T_b=lambda^2 S/(2 k Omega_b). At 45 arcsec, T_b=307.9 K. Doubling both beam axes multiplies Omega_b by 4, so T_b=76.98 K for the same 1 Jy/beam.
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: Thermal versus systematic floor
Worked solution: Thermal rms scales as tau^-1/2: 2.0/sqrt(4)=1.0 mJy at 4 h and 2.0/sqrt(16)=0.50 mJy at 16 h. Quadrature with 1.5 mJy gives sqrt(0.50^2+1.5^2)=1.581 mJy.
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.