Lab 06: Giant Planet Density and the Roche Limit at Uranus

Apparatus and Setup

This lab extends Lecture 11's giant-planet density comparison and Lecture 12's Roche-limit derivation to Uranus, using real Voyager 2 and stellar-occultation-derived ring data.

Materials and Data

Procedure

1. Compute Uranus's mean density from its mass and radius, and compare it to Jupiter's and Saturn's densities from Lecture 11.
2. Compute the fluid-body Roche limit for ice-density ring particles around Uranus, using the identical formula from Lecture 12.
3. Compare your computed Roche limit to the epsilon ring's real orbital radius (expressed in Uranus radii), and state whether the epsilon ring lies inside or outside your computed Roche limit.
4. Uranus's rings are known to be composed of unusually dark material, with lower albedo than Saturn's icy rings, suggesting a different (or processed) composition rather than pure water ice. Explain in one paragraph how this composition difference could affect the accuracy of your Roche-limit estimate from step 2, which assumed pure ice density.
5. Using the giant-planet interior concepts from Lecture 11, explain briefly why Uranus, despite being much less massive than Jupiter, still generates its own magnetic field (previewed in Lecture 10), even though its interior lacks a metallic-hydrogen layer.

Analysis and Uncertainty

The exact density of Uranus's ring particles is not independently well constrained; using pure ice density is a simplifying assumption whose effect on the computed Roche limit should be explicitly stated as a limitation, not hidden, following this course's standard for honestly characterizing approximations.

Deliverables

• Computed Uranus mean density, compared to Jupiter and Saturn.
• Computed fluid Roche limit for Uranus.
• A comparison to the real epsilon ring radius, correctly stating whether it lies inside or outside the Roche limit.
• Written paragraphs on ring-composition uncertainty and Uranus's magnetic field.

Assessment Criteria

• Correct density calculation and comparison.
• Correct Roche-limit calculation using the fluid-body formula.
• Correct, properly qualified comparison to the real ring radius.
• Correctly explains that Uranus's dynamo operates in an ionic water/ammonia mantle rather than metallic hydrogen.

References and Provenance

• OpenStax Astronomy 2e (local reference copy: references/openstax-astronomy-2e-extracted.txt), Section 11.4.
• Uranus's ring radii: standard published Voyager 2/stellar-occultation values (Elliot et al. 1977; Voyager Imaging Team 1986), not independently re-verified by live fetch this session; see materials/ASTR320/reference-log.md.
• Generated by materials/ASTR320/src/generate_astr320_labs_psets.py.