AND FURTHER ADVENTURES WITH R Orbital Similarity AND FURTHER ADVENTURES WITH R

Formation of Meteoroid streams Solid particles released from an active comets or NEAs Orbit remains similar to that of parent body: Ejected at very low velocity relative to the parent Resulting orbital dispersion is small Orbit dispersion will occur: Dynamic (chaotic) evolution of meteoroid stream Gravitational perturbations Radiation effects (Poynting—Robertson drag) Collisions Predominance of larger particles in older streams Dispersion increases over time until, eventually a stream will become dispersed into the background

ORBITAL SIMILARITY A fundamental question is whether a meteor belongs to a stream or is just part of the random background Establishing the degree of orbital similarity essential to: Associating meteors with known streams Associating meteors with parent bodies identification of new streams Knowing the origin of a stream gives us more clues to: Ejection process(es) Time of possible ejection(s) / age of stream Past and present orbits of parent bodies.

Orbital Similarity - Challenges Weak showers: identifying a ‘signal’ from the background Related objects may appear dissimilar due to Meteor stream evolution Uncertainties in measurements Unrelated meteors may appear to be similar by chance. Contamination from the background sporadic meteors

Stream Association Stream Identification

UFO software UFO uses coarse classification system using a radiant list: Date range Direction Velocity Accuracy improves with Unified observations We can play with the parameters / tolerances: Extension days Radiant diameter (%) Velocity (%)

HOW DO WE GET MORE SCIENTIFIC RIGOUR? Separating meteor showers from the sporadic meteor background is critical With UFO we have no means to understand / measure false positives

HOW STATISTICIANS WOULD APPROACH THIS ? Cluster analysis: sets of objects are grouped so that objects in the same group (called a cluster) are more similar to each other than to those in other groups (clusters). Grouping is usually based on a Distance function that gives a value to the distance between data points

Southworth and Hawkins (1963) First quantitative measure of orbital similarity First use of a computer to search for associations with parent body A form of cluster analysis

Southworth and Hawkins - DSH Uses analogy with a five-dimensional orthogonal coordinate system: Each orbital heliocentric element considered as a coordinate A meteoroid orbit is represented by a point in this co- ordinate space The distance between two points is a measure of the degree of similarity between two meteoroid orbits. Two orbits are associated if DSH is below a certain threshold

Calculating DSH I21 is the angle between the planes of each orbit Eccentricity (e) Inclination (i) Argument of periapsis (ω) Longitude of the ascending node (Ω) Perihelion distance (q) I21 is the angle between the planes of each orbit II21 is the angle between their respective perihelion points:

Modifications DD - DRUMMOND (1982) DD - DRUMMOND (1982) DH - JOPEK (1993) Use of chords as opposed to actual angles to represent I21 and II21 To address problems with large perihelion distances / eccentricities the first two terms of DSH are divided by the sum of the respective orbital elements in DD. Each term in DD is weighted to provide a dimensionless value in range 0 to 1. Takes features from both DSH and DD to create a further D- criterion (DH) VALSECCHI, JOPEK & FROESCHLE (1999). Based directly on comparison of the physical difference between the orbits

D-Criterion Calculations Using D-Criterion D-Criterion Calculations

Analysis Tools Excel UFORadiant UKMON R Suite

Excel

UFO Radiant (Sonotaco)

UKMON R-SUITE GITHUB REPOSOITORY ALGORITHMS Simple D-Analysis DSH (Southworth & Hawkins) DD (Drummond) DH (Jopek) Using SAS: Iterative analysis https://github.com/UKMeteorNetwork

Dd Calculated using J8_Per AS Reference UKMON data 2012 - 2017. Filters: EDMOND quality criteria J8_Per date range

Determining the D-Criterion Cut-off  

Determining the D-Criterion Cutoff Sporadic background begins to dominate Breakpoint at DD = 0.15 Outside / dispersed region of the stream Most Concentrated region of the stream. N changes considerably with D I Neslusan, J Svoren, V Porubcan (1995)

Dd Calculated using ALL UNIFIED OBS UKMON data 2012 - 2017. EDMOND quality criteria Full date range

Finding streams Hierarchical clustering (Southworth): Linking to nearest neighbour Accept Link if D below some criterion No knowledge of mean orbit is needed Iterative (Sekanina): Points inside a hypersphere of radius Dc centred on mean orbital elements Centre of sphere moves with each iteration Hypersphere: In geometry of higher dimensions, a hypersphere is the set of points at a constant distance from a given point called its centre.

ITERATIVE ANALYSIS No Yes Yes No Calculate Mean Orbit Calculate D-Criterion for all orbits against mean Any orbits with D > D0 Remove Orbit with highest D value Calculate Mean Orbit Calculate D-Criterion for all orbits against mean Any orbits with D > D0 Re-introduce Orbit with highest D value No Yes Yes No

Conclusions UFO Orbit does quite well. At DH = 0.15: False Positives < 7% Result is not impacted by date filter D-Criterion has its detractors Other approaches: Geocentric rather than Heliocentric view …?