Apparatus and Setup
Part 1 reproduces the Earth-to-Mars Hohmann transfer calculation from Lecture 11 in full unit-conversion detail. Part 2 computes the Sun-Earth and Sun-Jupiter L1 distances and evaluates the Alpha Centauri AB-Proxima hierarchy ratio from Lecture 12.
Materials and Data
| Quantity | Value |
|---|---|
| Earth semimajor axis, eccentricity | 1.0000 AU, 0.0167 |
| Mars semimajor axis | 1.5237 AU |
| Sun standard gravitational parameter GM_Sun | 1.3275e+20 m³/s² |
| Jupiter mass | 1.898e+27 kg |
| Alpha Cen AB semimajor axis | 23.299 AU |
| Proxima periastron distance about AB | 4100 AU |
Procedure
- Compute the Hohmann transfer orbit's semimajor axis (average of Earth's and Mars's semimajor axes), then compute Earth's circular-orbit speed, the transfer orbit's perihelion speed, and the first burn Δv1.
- Compute Mars's circular-orbit speed, the transfer orbit's aphelion speed, and the second burn Δv2, then the total mission Δv (sum of both burns) and the one-way transfer time in days.
- Compute the Sun-Earth L1 distance using r_L1 ≈ R(m_Earth/3m_Sun)^(1/3), and separately compute the Sun-Jupiter L1 distance using the same formula with Jupiter's mass and orbital radius (5.2 AU); compare the two L1 distances and explain why Jupiter's is so much larger despite Jupiter being much farther from the Sun.
- Compute the ratio of Proxima Centauri's periastron distance to the Alpha Centauri AB semimajor axis, and state whether this hierarchy ratio is consistent with published long-term stability studies of the Alpha Centauri triple system.
Analysis and Uncertainty
The Hohmann transfer calculation assumes perfectly circular, coplanar orbits for both Earth and Mars; state explicitly which of the two assumptions (circular, coplanar) is least accurate for the real Earth-Mars system (recall Mars's actual eccentricity is about 0.093, ASTR120/ASTR320 territory) and what qualitative effect relaxing that assumption would have on the required Δv.
Deliverables
- Full Hohmann transfer calculation (both burns, total Δv, transfer time).
- Sun-Earth and Sun-Jupiter L1 distance calculations with a written comparison.
- Alpha Centauri AB-Proxima hierarchy-ratio calculation and stability discussion.
Assessment Criteria
- Correct vis-viva arithmetic at all four orbital points (Earth circular, transfer perihelion, transfer aphelion, Mars circular).
- Correct transfer-time calculation using Kepler's third law in AU/yr units.
- Correct L1 distance calculations for both Sun-Earth and Sun-Jupiter, with correct mass ratios.
- Correct hierarchy-ratio arithmetic and a physically sound stability discussion.
References and Provenance
- OpenStax Astronomy 2e (local reference copy: references/openstax-astronomy-2e-extracted.txt), Chapter 3.4-3.6.
- Alpha Centauri AB and Proxima orbital elements: live-verified this session.