1. Transfer orbit setup
a_transfer = 0.5(1.0000 + 0.7233) = 0.8617 AU. Perihelion = 0.7233 AU (Venus's orbit), aphelion = 1.0000 AU (Earth's orbit).
| Criterion | Points |
|---|---|
| Correct transfer semimajor axis | 8 |
| Correct identification of perihelion/aphelion | 7 |
Worked Solutions
2. Both burns
Δv1 (leaving Earth, at transfer aphelion) = |29.79 − 27.29| = 2.50 km/s. Δv2 (arriving at Venus, at transfer perihelion) = |35.03 − 37.73| = 2.71 km/s. Total Δv = 5.20 km/s.
| Criterion | Points |
|---|---|
| Correct vis-viva speeds at both transfer points | 12 |
| Correct Δv arithmetic and total | 8 |
3. Transfer time
t = 0.5 × a_transfer^1.5 = 0.400 yr = 146 days, shorter than the Earth-Mars transfer time (259 days) by a ratio of 1.77, because the Venus transfer orbit has a smaller semimajor axis and Kepler's third law gives shorter periods for smaller orbits.
| Criterion | Points |
|---|---|
| Correct transfer-time arithmetic | 8 |
| Correct comparison and explanation | 7 |
4. Sun-Jupiter L1
r_L1(Sun-Jupiter) = 5.2044 AU × (1.898e+27/(3×1.989e+30))^(1/3) × (AU in km) = 5.315e+07 km, versus 1.496e+06 km for Sun-Earth, a ratio of 35.5. This is much larger than the orbital-radius ratio (5.20) alone because r_L1 also scales with (mass ratio)^(1/3), and Jupiter's mass ratio to the Sun is far larger than Earth's, so both factors increase the Jupiter L1 distance relative to a naive orbital-radius-only scaling.
| Criterion | Points |
|---|---|
| Correct L1 arithmetic for Sun-Jupiter | 12 |
| Correct explanation combining both R and mass-ratio scaling | 8 |
Grading Notes
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 (AU/m, years/seconds, km/Mpc, Msun/kg) -- these are the most common source of order-of-magnitude mistakes in this course.