1. AND FURTHER ADVENTURES WITH R
Orbital Similarity
AND FURTHER ADVENTURES WITH R

2. 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

3. 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.

4. 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

5. Stream Association Stream Identification

6. 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 (%)

7. 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

8. 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

9. 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

10. 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

11. 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:

12. 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

13. D-Criterion Calculations
Using D-Criterion
D-Criterion Calculations

14. Analysis Tools
Excel
UFORadiant
UKMON R Suite

15. Excel

16. UFO Radiant (Sonotaco)

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

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

19. Determining the D-Criterion Cut-off

20. 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)

21. Dd Calculated using ALL UNIFIED OBS
UKMON data
EDMOND quality criteria
Full date range

22. 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.

23. 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

24. 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 …?