A Close Moon Around a Trans-Neptunian Object
An 18-month astrometric campaign detects a moon around a trans-Neptunian object. The moon's projected separation is 2,400 km, and its orbital period is 1.60 days, measured with 5% uncertainty. The primary's effective radius is 400 km, and the moon's radius is 110 km. The visible geometric albedo is 0.08. A colleague argues that the short orbital period proves the system is a dense rocky body, perhaps a collisional remnant, because icy bodies cannot hold such a close moon. You also have two nights of new astrometry that show the moon's position but not a full orbit. The team wants to know whether the mass and density support the rocky-body claim, and what assumptions dominate the error budget. No other satellites or rings are detected; the object is at 40 AU, so observations are photon-limited.
1. Compute the total system mass from the quoted separation and period. State the formula, show the unit conversions, and give the result in kg.
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Use Kepler's third law for a circular orbit: M_total = 4π² a³ / (G P²). Convert P = 1.60 d × 86,400 s/d = 1.3824×10^5 s, so P² = 1.911×10^10 s². Convert a = 2,400 km = 2.4×10^6 m, so a³ = 1.382×10^19 m³. Then M = 39.48 × 1.382×10^19 / (6.674×10^-11 × 1.911×10^10) = 5.46×10^20 / 1.275 = 4.3×10^20 kg. The result is the total mass, not the primary mass; the moon contributes little if its density is similar, but the primary dominates. This is roughly 0.07 Earth masses, or about 7% of Pluto's mass. The calculation assumes the quoted separation is the semimajor axis and the orbit is circular; both assumptions need checking before overinterpreting the density.
2. Estimate the bulk density using the two radii and judge the colleague's claim that the system is a dense rocky body. What caveats matter?
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Volumes: V_primary = (4/3)π(4.0×10^5 m)^3 = 2.68×10^17 m³; V_moon = (4/3)π(1.10×10^5 m)^3 = 5.6×10^15 m³; total ≈ 2.74×10^17 m³. Density = 4.28×10^20 kg / 2.74×10^17 m³ ≈ 1.56×10^3 kg/m³, or 1.56 g/cm³. That is far below rocky densities (roughly 3–5 g/cm³) and close to ice-rich or porous material, so the colleague's claim is not supported. A density near 1.6 g/cm³ is consistent with a mixture of rock and water ice, or a porous rubble pile. Caveats: effective radii may include unresolved topography; the moon's mass is counted in the total but not separated; and if the true semimajor axis is larger than the projected separation, the mass and density rise. Even so, reaching rock density would require a large, unlikely correction.
3. The period has a 5% uncertainty. Which other assumption is likely to bias the mass more: the 5% period error, the circular-orbit assumption, or the single-epoch projected separation? Explain.
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The mass scales as M ∝ a³ P^−2. A 5% period error alone gives about 10% mass error, because fractional error is doubled. A 10% separation error would give about 30% mass error, so separation dominates. The single-epoch projected separation is the larger concern: unless 2,400 km is the maximum projected separation, the true semimajor axis is larger, so 4.3×10^20 kg is a lower limit. The circular assumption also matters: for an eccentric orbit, the instantaneous separation is not a, and using it can bias the mass either way. The two nights of astrometry are not enough to fit an orbit; a full orbit would constrain inclination and eccentricity. Thus the period error is the least important of the three for this claim.