Jupiter Trojans with strict Saturn resonance

For this preliminary analysis, I have utilized current osculating elements. While a definitive dynamical study would require proper elements to account for long-term averages, this high-precision "snapshot" serves as a first step to reveal the instantaneous resonant architecture of the system.I defined two specific cohorts based on the ratio of the asteroid’s orbital period :

$P_{as}$

• Jovian Mode (r1:1): The fundamental frequency of the Jupiter Trojan clouds.

• Saturnian Mode (r5:2): The sub-harmonic frequency associated with the Saturnian "Great Inequality."

An asteroid is "flagged" for a cohort only if it meets a strict relative tolerance:

$$\mid \frac{P_{ast}\mathrm{/}{P}_{planet}}{r}-1\mid <ϵ$$

This filter isolates only the peak of the resonant signal, effectively decoupling the cloud into objects strictly synchronized with Jupiter's frequency and those whose orbits are currently modulated by the Saturnian 5:2 harmonic.

Why this strictness matters
By narrowing the tolerance to 0.001, I am essentially creating a "Dynamical Spectrometer." This isolates only the "peak" of the resonant signal, filtering out the background noise of objects that are merely passing through the neighborhood of a resonance.

This provides the "lens" necessary to decouple the Jupiter Trojan clouds into two distinct populations:

• Objects strictly synchronized with Jupiter's 1:1 frequency ($NumObjects=713)$

  
  

• Objects whose periods are currently aligned with the Saturnian 5:2 frequency ($NumObjects=626$

Structural Analysis: Longitudinal Depletion Zones

Next, I investigated the "texture" of these populations by analyzing the distribution of their longitudes of perihelion 

$\varpi =\omega +\mathrm{\Omega }$

Using an Adaptive Gap-Searching Algorithm, I identified the longitudinal structure of these groups.

Rather than assuming these gaps are permanent "ejection zones," I define them as Longitudinal Depletion Zones—sectors of phase space that are currently under-populated or "forbidden" at this specific epoch.

The algorithm identifies two types of structures:

• Islands (Occupied Sectors): Longitudinal ranges where asteroids cluster together, suggesting pockets of relative stability or "safe harbors."

• Gaps (Depletion Zones): Longitudinal ranges where no asteroids were found in the current census. These likely represent the "porosity" of the gravitational field.

Details

Jupiter Resonace r1_1

   planet resonance            Type     start       end      width n_ast n_com n_total   density threshold
1 Jupiter      r1_1     EMPTY (Gap) -157.4658 -150.6931   6.772678     0     0       0 0.0000000         5
2 Jupiter      r1_1 FILLED (Island) -150.6931 -136.8178  13.875267     0    12      12 0.8648482         5
3 Jupiter      r1_1     EMPTY (Gap) -136.8178 -129.3879   7.429940     0     0       0 0.0000000         5
4 Jupiter      r1_1 FILLED (Island) -129.3879  148.2427 277.630610     0   660     660 2.3772595         5
5 Jupiter      r1_1     EMPTY (Gap)  148.2427  154.4604   6.217691     0     0       0 0.0000000         5
6 Jupiter      r1_1 FILLED (Island)  154.4604  167.2567  12.796277     0    11      11 0.8596250         5
7 Jupiter      r1_1     EMPTY (Gap)  167.2567  172.4148   5.158141     0     0       0 0.0000000         5
8 Jupiter      r1_1 FILLED (Island)  172.4148 -157.4658  30.119396     0    30      30 0.9960359         5

9         5

Saturn Resonace r5_2

  planet resonance            Type     start       end      width n_ast n_com n_total   density threshold
1 Saturn      r5_2     EMPTY (Gap) -153.2552 -146.1287   7.126455     0     0       0 0.0000000     4.166
2 Saturn      r5_2 FILLED (Island) -146.1287 -138.1086   8.020117     0     4       4 0.4987458     4.166
3 Saturn      r5_2     EMPTY (Gap) -138.1086 -133.9423   4.166236     0     0       0 0.0000000     4.166
4 Saturn      r5_2 FILLED (Island) -133.9423 -153.2552 340.687192     0   622     622 1.8257217     4.166

Column meaning:

• Type:

  • FILLED (Island): A longitudinal range where asteroids are "clumping" together. These represent stable pockets where orbits can safely exist.

  • EMPTY (Gap): A "Forbidden Corridor." A longitudinal range where not a single asteroid was found in the current census, indicating a likely unstable or "ejection" zone.

• start / end: The boundaries of the band in degrees, measured in Longitude of Perihelion (

  $\varpi =\omega +\mathrm{\Omega }$

  
  

  • Values are wrapped to the standard 

    $[-{180}^{\circ },{180}^{\circ }]$

     range.

  • Note: If an island's start is a large positive number and end is a large negative number (e.g., 

    ${179}^{\circ }$

    to 

    $-{178}^{\circ }$

    ), it means the island straddles the 180-degree "seam" of the coordinate system.

• width: The angular span of the band in degrees.

  • For Islands, this measures the "Libration Width" (how much an asteroid can wobble and still remain in that specific clump).

  • For Gaps, this measures the size of the "Forbidden Corridor."

• n_ast / n_com: The census count of Asteroid-like (

  $T_j\ge 3.0$

  ) and Comet-like (

  $T_j<3.0$

  ) objects within that specific island. (Naturally, these are 0 for EMPTY rows).

• n_total: The total number of objects in that band (

  $n_{ast}+n_{com}$

  ).

• density: Calculated as 

  $n_{total}\mathrm{/}width$

  . This represents the "Population Packing" of the island.

  • A high density indicates a "Hyper-Stable" core where many asteroids are squeezed into a tiny longitudinal slot.

  • A low density indicates a sparse or "Diffuse" population.

• threshold: The Structural Grain (in degrees) used for the analysis. This is the "chain-breaker" value: any empty space wider than this threshold was officially declared an EMPTY (Gap), splitting the population into separate islands. The threshold is determined by the algorithm: we determine this value by scanning different sizes until we find a Stability Window — a 'elbow' where the number of clusters doesn't change.

Notes:

If this census is correct, there is a disparity in how these two planets "shatter" the Trojan population.

• Jupiter: being the primary host of the 1:1 resonance, Jupiter’s influence is characterized by greater fragmentation. It carves out a higher number of discrete islands separated by broad, "violent" gaps averaging 6.39° in width. This suggests that the local primary mass acts as a broad-spectrum filter, clearing wide swathes of phase space where orbits cannot persist. The primary perturber clears a wider path.
• Saturn: in contrast, the Saturnian 5:2 mode exhibits a much more contiguous and "saturated" structure. It possesses fewer islands and significantly narrower forbidden corridors, averaging  5.65°. This is consistent with it being a secondary perturber.

We are measuring a 1.13 x difference in the "structural grain" of the two modes. While Jupiter sets the global boundaries of the Trojan clouds, Saturn’s harmonic interaction provides a higher-resolution "comb" that allows for a more tightly packed and continuous distribution of objects.

This suggests that the "Maximum Void Width" is a direct indicator of gravitational intensity: the more massive and closer the perturber, the wider the "depletion zones" it creates.

While the fragmentation (number of islands) and porosity (width of gaps) differ between the two modes, both the Jovian 1:1 and Saturnian 5:2 populations are dominated by a single "Mainland" island that hosts the vast majority of the respective asteroids.

When we calculate the midpoint (mean longitude) of these mainland islands, the result is as follows:

• Jupiter r1_1 Mainland (N=660/713 --> 92%):
• Span: -129.39° to 148.24°
• Midpoint: ~9.4°
• Saturn r5_2 Mainland (N=622/626 -->99%):
• Span: -133.94° to -153.25° (wrapping around the 180° seam)
• Midpoint: ~36.4

The current Longitude of Perihelion for Jupiter is approximately 15.0°.

The current Longitude of Perihelion for Saturn is approximately 92.0°.

Midpoint Alignment

$\mathrm{<span\; style="font-family:\; courier;">\varpi </span>}$

• The Jovian Mainland midpoint (~9.4°) being relatively near Jupiter’s own perihelion (15.0°) is exactly what celestial mechanics predicts. Most Trojans have their perihelia "locked" or oscillating around Jupiter's perihelion to maintain secular stability.

• The Saturnian Mainland midpoint (~36.4°) is fascinating. It suggests that objects most sensitive to the Saturn 5:2 resonance are shifted roughly 20-25° away from the Jovian average towards the Saturn perihelium. These asteroids are "tilting" their orbits to find a gravitational equilibrium between the two giants.

The "Anti-Alignment" Void

There is a striking correlation in the location of the Gaps:

• Jupiter Gap 1: -157.4° to -150.6°

• Saturn Gap 1: -153.2° to -146.1°

These two gaps almost perfectly overlap. The midpoint of these gaps is roughly -152°
Jupiter's perihelion is at 15°. The "Anti-perihelion" (aphelion direction) is -165°.
There is an Instability Corridor near the anti-alignment of Jupiter’s orbit. Asteroids whose perihelia drift into this sector likely experience a "kick" that pushes them out.

 

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