Background Techniques Results Conclusions Resonant Origins for Pluto’s High Inclination Curran D. Muhlberger University of Maryland, College Park April 7, 2008 Curran D. Muhlberger Resonant Origins for Pluto’s High Inclination
Background Introduction Techniques Planetary Migration Results Orbital Resonances Conclusions Goals Explain Pluto’s high eccentricity ( e = 0 . 24 ) and high inclination ( i = 17° ) using resonances Three candidates 6:4 mean motion resonance 1 1:1 secular resonance 2 2:1 secular resonance 3 Curran D. Muhlberger Resonant Origins for Pluto’s High Inclination
Background Introduction Techniques Planetary Migration Results Orbital Resonances Conclusions Planetary Migration by Scattering Planetesimals Planets other than Jupiter preferentially scattered planetesimals inward, migrated outward Migrations move locations of resonances, catching Pluto If migration rate is slow enough, characteristic effect on resonances is rate-independent -0.2 1 5 8 Curran D. Muhlberger Resonant Origins for Pluto’s High Inclination
Background Introduction Techniques Planetary Migration Results Orbital Resonances Conclusions Orbital Elements & Symmetries Orbital Elements: a , e , i , Ω , Celestial body ϖ = Ω + ω , λ ( ˙ λ ≈ n ) Secular Variables True anomaly ν h = e sin ( ϖ ) Argument of periapsis ω Ω k = e cos ( ϖ ) ♈ Longitude of ascending node Reference direction p = sin ( i / 2 ) sin ( Ω ) Plane of reference i Inclination q = sin ( i / 2 ) cos ( Ω ) ☊ Ascending node t i b O r Eigenfrequencies: f , g Curran D. Muhlberger Resonant Origins for Pluto’s High Inclination
Background Introduction Techniques Planetary Migration Results Orbital Resonances Conclusions Resonant Behavior Mean Motion Resonance Secular Resonance Simple ratio of orbital periods Simple ratio of precession (dependent on λ , n ) periods (averaged orbits) Form resonant arguments subject to symmetries Good: 6 λ P − 4 λ N − 2 Ω P , 2 Ω P − Ω N − Ω J Bad: 3 λ P − 2 λ N − Ω N , 2 Ω P − Ω N Capture Jump Curran D. Muhlberger Resonant Origins for Pluto’s High Inclination
Background Techniques Numerical Methods Results Conclusions Simulation and Analysis Both of pre-existing and new software used throughout project. Used HNBody and HNDrag to simulate Solar System over billions of years ( > 24GB of data generated) To determine secular eigenfrequencies, wrote code to perform FFT on orbital elements Features of PowerSpectrumEstimator : Data windowing to reduce spectral leakage Overlapping data segments to minimize variance Automatic peak finding with inverse quadratic interpolation Removal of aliased peaks Orthogonality of total angular momentum Curran D. Muhlberger Resonant Origins for Pluto’s High Inclination
Background Techniques Numerical Methods Results Conclusions Example: Outer Solar System p Spectra Matches g 6 to better than 1% ; matches g 7 to within 7% ; matches g 8 to within 25% ; g 5 is effectively 0 Spectra of 'p' for the Outer Solar System 6.727809e-09 Jupiter 1.818169e-09 5.609270e-08 Saturn 3.390916e-11 Uranus Neptune 0 2e-08 4e-08 6e-08 8e-08 1e-07 Frequency [cycles/day] Curran D. Muhlberger Resonant Origins for Pluto’s High Inclination
Background Mean Motion Resonances Techniques Secular Resonances (1:1) Results Secular Resonances (2:1) Conclusions Candidate #1 – 6:4 Mean Motion Resonance Pluto is currently trapped in a 3:2 eccentricity resonance ϖ P ) and a Kozai resonance ( ˙ ( 3 n P − 2 n N − ˙ Ω P − ˙ ϖ P ). Together, these imply a 6:4 inclination resonance ( 6 n P − 2 n N − 2 ˙ Ω P ). Initially, these were split (no Kozai resonance) Being first-order, eccentricity resonance is stronger Simulations rule out capturing in inclination resonance first What about afterwards? Curran D. Muhlberger Resonant Origins for Pluto’s High Inclination
Background Mean Motion Resonances Techniques Secular Resonances (1:1) Results Secular Resonances (2:1) Conclusions Examples of Mean Motion Resonances Orbital Elements for Pluto Orbital Elements for Pluto 33.004 33.12 33.1 33.002 33.08 33 33.06 a [AU] 32.998 a [AU] 33.04 32.996 33.02 33 32.994 32.98 32.992 32.96 32.99 32.94 0.0014 0.035 0.0012 0.03 0.001 0.025 0.0008 0.02 e e 0.0006 0.015 0.0004 0.01 0.0002 0.005 0 0 0.000102 1.6 1.4 0.0001015 1.2 0.000101 1 i [deg] 0.0001005 0.8 0.6 0.0001 0.4 9.95e-05 0.2 9.9e-05 0 0 5e+07 1e+08 1.5e+08 2e+08 0 1e+09 2e+09 3e+09 4e+09 5e+09 6e+09 7e+09 8e+09 t [yr] t [yr] Migration rates too slow, inclination rise too small Curran D. Muhlberger Resonant Origins for Pluto’s High Inclination
Background Mean Motion Resonances Techniques Secular Resonances (1:1) Results Secular Resonances (2:1) Conclusions Candidate #2 – 1:1 Secular Resonance A 1:1 resonance ( ˙ Ω P − ˙ Ω N ) should be easier to find and more powerful than a 2:1 resonance. Studied an idealized Jupiter+Neptune+Pluto system May have been present at Solar System formation Could capture into 3:2 mean motion resonance at just the right time, maintain high inclination Curran D. Muhlberger Resonant Origins for Pluto’s High Inclination
Background Mean Motion Resonances Techniques Secular Resonances (1:1) Results Secular Resonances (2:1) Conclusions Example of 1:1 Secular Resonance Plot and Spectrum of Pluto's 'p' Orbital Elements for Pluto 33.6 Pluto 33.5 Neptune 33.4 a [AU] 33.3 33.2 33.1 33 32.9 0.018 0.016 0.014 0.012 0.01 e 0.008 0.006 a P = 33 AU 0.004 0.002 0 5e-07 1e-06 1.5e-06 2e-06 0 0.25 30 0.2 25 0.15 0.1 20 0.05 i [deg] 0 15 -0.05 10 -0.1 -0.15 5 -0.2 -0.25 0 0 2e+07 4e+07 6e+07 8e+07 1e+08 1.2e+08 0 2e+07 4e+07 6e+07 8e+07 1e+08 1.2e+08 Time [yr] Time [yr] Static inclination resonance extremely broad and powerful (3 AU, 25° ) Curran D. Muhlberger Resonant Origins for Pluto’s High Inclination
Background Mean Motion Resonances Techniques Secular Resonances (1:1) Results Secular Resonances (2:1) Conclusions Secular Resonances in the Solar System In full Solar System, 1:1 resonance is not as broad or powerful. Still, migrating across makes jump or capture possible. Inclination jump of 10°observed near initial conditions Capture raises more questions: when/how did it break out? Leaves observed 2:1 resonance a coincidence Early proximity to 1:1 indicates that 2:1 was not active prior to capture in eccentricity resonance. Curran D. Muhlberger Resonant Origins for Pluto’s High Inclination
Background Mean Motion Resonances Techniques Secular Resonances (1:1) Results Secular Resonances (2:1) Conclusions Candidate #3 – 2:1 Secular Resonance By raising M U → 1 . 8 M U , we could create conditions where 2 p 1 ≈ g 8 . By dragging Pluto directly, we could study strength of jump and capture. Raising M U ⇐ ⇒ increasing Uranus’s initial position Jump is too weak (2° ) to explain current inclination What about capture? Curran D. Muhlberger Resonant Origins for Pluto’s High Inclination
Background Mean Motion Resonances Techniques Secular Resonances (1:1) Results Secular Resonances (2:1) Conclusions Example of 2:1 Capture Spectra of 'p' for the Outer Solar System: Initial Orbital Elements for Pluto 38.3 Jupiter 38.2 3.874448e-07 Saturn 38.1 7.669366e-07 Uranus 38 Neptune a [AU] 37.9 Pluto 37.8 37.7 37.6 37.5 37.4 0.24 0.235 0.23 0.225 0.22 e 0.215 Spectra of 'p' for the Outer Solar System: Resonance (2e7 yr) 0.21 0.205 Jupiter 0.2 4.129240e-07 Saturn 0.195 7.669416e-07 Uranus Neptune 16 Pluto 14 12 10 i [deg] 8 6 4 2 0 0 5e-07 1e-06 1.5e-06 2e-06 2.5e-06 3e-06 3.5e-06 4e-06 0 2e+07 4e+07 6e+07 8e+07 1e+08 1.2e+08 1.4e+08 1.6e+08 1.8e+08 Frequency / 2 π [rad/yr] t [yr] Capture is possible! Yields i → 16° + Curran D. Muhlberger Resonant Origins for Pluto’s High Inclination
Background Mean Motion Resonances Techniques Secular Resonances (1:1) Results Secular Resonances (2:1) Conclusions Active Resonances in 2:1 Capture 3 λ P - 2 λ N - ϖ P 2 Ω P - Ω N - 0 350 350 300 300 250 250 200 200 150 150 100 100 50 50 0 0 6 λ P - 4 λ N - 2 Ω P 2 Ω P - 2 ϖ P 350 350 300 300 250 250 200 200 150 150 100 100 50 50 0 0 Curran D. Muhlberger Resonant Origins for Pluto’s High Inclination
Background Techniques Results Conclusions Summary Currently, no overwhelmingly likely explanation. However, some can be ruled out while others can be constrained. Resonance Grade Pros Cons Mean Motion D Currently active Could not capture Too weak Secular 1:1 B Strong enough Not active today Possibly active in early solar system Large jump instead of capture Secular 2:1 B+ Possibly active today M U → 1 . 8 M U Capable of capture Dragging Pluto, not Neptune Curran D. Muhlberger Resonant Origins for Pluto’s High Inclination
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