Shocks, Cosmic Rays, and Magnetic Fields
T2(Tycho)=2.93e+08 K versus T2(Cas A)=3.46e+08 K -- Cas A's shock produces the hotter predicted plasma, by 18.1%, directly reflecting its somewhat higher shock velocity through the T2 ∝ v_shock^2 scaling.
| Criterion | Points |
|---|---|
| Correct post-shock temperature arithmetic for Tycho | 8 |
| Correct percentage comparison | 7 |
Above the knee (index 3.1): flux ratio over a factor-of-10 energy increase = 10^(-3.1)=7.94e-04. Below the knee (index 2.7): 10^(-2.7)=2.00e-03 -- the above-knee decline is steeper (a smaller ratio), consistent with the steeper power-law index above the knee.
| Criterion | Points |
|---|---|
| Correct power-law ratio arithmetic for both indices | 8 |
| Correct comparison of steepness | 7 |
B(Barnard 68 density)=377.5 microG versus B(OMC-1 core density)=436 microG -- OMC-1's denser core is predicted to have the stronger field, consistent with the Crutcher (2012) B ∝ n^0.65 scaling above the critical density.
| Criterion | Points |
|---|---|
| Correct B-field arithmetic for Barnard 68 | 8 |
| Correct comparison and scaling justification | 7 |
nu_c(gamma=1.0e4)=8.398e+10 Hz, in the radio regime -- consistent with the nu_c ∝ gamma^2 scaling, since a factor-of-100 decrease in gamma (from 1.0e6 to 1.0e4) should decrease nu_c by a factor of 100^2=10,000, moving the emission from optical (lab's gamma=1.0e6 case) down to radio frequencies.
| Criterion | Points |
|---|---|
| Correct synchrotron-frequency arithmetic | 8 |
| Correct identification of the radio regime and the gamma^2 scaling | 7 |
Award full marks for correct method, units, and a numeric result consistent with the arithmetic shown (allow reasonable rounding). Verify that the student re-derives the number for this problem's specific inputs rather than quoting a lecture example without adaptation. Watch specifically for unit-conversion errors (cm^-3 vs m^-3, pc vs m, K km/s integrals), for comparative-magnitude statements given in the wrong direction, and for level-1/level-2/level-3 data-provenance conflation (e.g., presenting the illustrative synthetic HI spectrum shape as an exact measurement of a specific real sightline) -- these are the most common sources of mistakes in this course.