Worked Solutions
1. Density and Composition of Pluto
ρ = M/(4/3πR³) = 1.303e+22 / [(4/3)π(1.1883e+06)³] = 1854 kg/m³. This is well below any solid rocky-planet density (>3900 kg/m³) and comparable to the icy-satellite range (1000-2000 kg/m³), consistent with Pluto being a Kuiper Belt object built largely from ices condensed far beyond the frost line, as the condensation sequence predicts.
| Criterion | Points |
|---|
| Correct density calculation with unit conversion | 6 |
| Correct comparison to icy-body density range | 5 |
| Correct qualitative conclusion tied to condensation sequence | 4 |
2. Titan's Density and the Ice/Rock Mixture
ρ = 1.3452e+23 / [(4/3)π(2.5747e+06)³] = 1882 kg/m³. A simple linear mixing estimate, x·3000 + (1-x)·917 = 1882, gives x ≈ 0.46, i.e., roughly 46% rock by mass. This estimate is only approximate because real planetary ices and rock do not simply average their densities when mixed (compression, porosity, and phase changes at Titan's actual internal pressures all modify the true mixed density), and because Titan is now known (from Cassini gravity data) to be at least partially differentiated rather than a uniform mixture.
| Criterion | Points |
|---|
| Correct density calculation | 5 |
| Correct linear-mixing rock-fraction estimate | 6 |
| Correctly identifies a specific limitation of the simple mixing model | 4 |
3. Frost Line for a Fainter and a Brighter Young Star
r = √[L/(16πσT⁴)], so r ∝ √L. For L = 0.5 L⊙: r = 2.43 AU. For L = 2 L⊙: r = 4.87 AU. Compared to the Sun's own frost line (∼2.8 AU), the frost line moves inward for a fainter young star and outward for a brighter one, scaling as the square root of luminosity.
| Criterion | Points |
|---|
| Correct application of the frost-line formula to both cases | 8 |
| Correctly states the sqrt(L) scaling | 7 |
4. Runaway versus Oligarchic Growth: A Conceptual Comparison
(a) Runaway growth is self-reinforcing because an embryo's effective gravitational-focusing cross-section grows with its own escape velocity relative to the surrounding planetesimals' velocity dispersion (cross-section ∝ 1 + (v_esc/v_rel)²), so a slightly larger embryo accretes disproportionately faster, widening the gap with its neighbors. (b) Oligarchic growth slows this process once a handful of embryos dominate their local feeding zones, because each embryo's own gravity begins to stir up (increase the velocity dispersion of) the remaining planetesimals in its zone, reducing the gravitational-focusing advantage and roughly equalizing growth rates among the surviving embryos.
| Criterion | Points |
|---|
| Correctly explains runaway growth via gravitational focusing (not just size) | 8 |
| Correctly explains the transition to oligarchic growth via self-stirring | 7 |
5. Critical Core Mass and the Gas Giant/Ice Giant Divide
Once a core exceeds the critical mass (∼5-10 Earth masses), runaway gas accretion can capture hundreds of Earth masses of hydrogen/helium within ∼10⁴-10⁵ years, but only if the gas disk is still present. If Jupiter's and Saturn's cores reached this threshold with several Myr of disk lifetime remaining, they could accrete gas for an extended period, explaining their ∼90%+ hydrogen/helium composition. If Uranus's and Neptune's cores formed more slowly (plausible given the much lower disk density at 19-30 AU) and only reached, or never fully reached, the critical mass shortly before the gas disk dispersed, they would have captured only a comparatively small hydrogen/helium envelope before the gas supply ran out, leaving them dominated by their icy/rocky cores -- exactly the observed ∼10-20% hydrogen/helium mass fraction.
| Criterion | Points |
|---|
| Correctly explains the critical-mass/runaway-accretion mechanism | 7 |
| Correctly connects accretion timing to the observed compositional divide | 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, days or years/seconds, bar/Pa) and for comparative-magnitude statements stated in the wrong direction (e.g., a tidal acceleration or greenhouse excess described as a multiple when it is actually a small fraction, or vice versa) -- these are the most common sources of silent error in this course.