Problem 2.1: worked solution
For a circular test-particle orbit, GM_sun/r^2=v^2/r and v=2*pi*r/P. Therefore P=2*pi*sqrt(r^3/GM_sun). With r=1 AU and Astropy's GM_sun converted to AU^3/day^2, P=365.256898 days. Using G and M separately gives the same expression when units are consistent; retaining Earth mass changes the central parameter to G(M_sun+M_earth) and is a negligible but definable correction. The check is dimensional: r^3/GM has units of time squared. The value is a circular two-body period, not a claim that the observed tropical year is exactly this duration.
Problem 2.2: worked solution
Initialize from the Jan 1 target-399 state and propagate to the Jan 2 TDB epoch in the recorded J2000 ecliptic, AU/AU-day convention. The REBOUND Sun-relative Newtonian result differs from the Jan 2 vector by 126.014 km; independent DOP853 differs by 126.079 km. Both compare simplified Sun-Earth integrations with a DE441 ephemeris state, so the residual includes omitted perturbations and frame/center differences; it is not a pure integration error. Refine the WHFast timestep and DOP853 tolerance, and transform both states into one frame before attributing the remaining residual. An acceptable alternative is to compare a transformed velocity state as well, provided the frame and epoch conversion is explicit. Source values are level-2, not live-verified this session.
Problem 2.3: worked solution
The generated circular-orbit sweep is:
| steps/orbit | dt (day) | relative energy error | one-period endpoint error (AU) |
|---|---|---|---|
| 20 | 18.2628 | 6.98e-16 | 9.436e-06 |
| 40 | 9.13142 | 1.629e-15 | 9.436e-06 |
| 80 | 4.56571 | 6.98e-16 | 9.436e-06 |
| 160 | 2.28286 | 1.745e-15 | 9.436e-06 |
Choose the coarsest step whose relevant endpoint/invariant errors meet a declared science tolerance; energy conservation alone does not bound phase/position error. The sequence should be checked for an asymptotic trend before extrapolating. A direct comparison with an analytic Kepler state is an acceptable independent test; state whether the target is phase accuracy, energy behavior, or both.
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.