I refer to this MPML message and to the associated conversation, in particular to the interesting comments made by Aldo Vitagliano, author of Solex orbit simulator, about the difficulty to see a cluster of KBOs.

I do not know the answer but this prompted me to look at the KBO's orbit parameter distribution. What follows next is an exercise ... so I do not claim that it is correct!

I used the Web service made available by MPC to look for KBOs characterized by:

250 AU < a < 1000 AU

I found this list:

Asteroid | a | e | i | w | om |
---|---|---|---|---|---|

2006 UL321 | 260.7642 | 0.9099041 | 37.36673 | -6.20276 | -17.0798 |

2012 VP113 | 268.2509 | 0.7005635 | 24.0183 | -66.968 | 90.88303 |

1996 PW | 271.4848 | 0.9905821 | 29.72461 | -178.15845 | 144.61041 |

2011 OR17 | 287.1875 | 0.9892086 | 110.33772 | 14.03442 | -88.40106 |

336756 | 319.0439 | 0.9704821 | 140.81617 | 132.96423 | 136.19752 |

2013 RF98 | 325.1039 | 0.8883746 | 29.57957 | -43.45499 | 67.57105 |

2004 VN112 | 339.0837 | 0.8604115 | 25.51725 | -32.76479 | 66.06033 |

2010 GB174 | 364.8513 | 0.8670264 | 21.53648 | -12.61199 | 130.60394 |

2015 DB216 | 418.7213 | 0.9800038 | 40.48768 | -119.1578 | 3.27749 |

2010 BK118 | 484.3816 | 0.987395 | 143.90195 | 179.11098 | 175.97502 |

2007 DA61 | 518.3954 | 0.9948766 | 76.71542 | -10.32894 | 145.98047 |

90377 | 539.668 | 0.8588315 | 11.92852 | -48.91693 | 144.50447 |

2007 TG422 | 546.4473 | 0.934885 | 18.57957 | -74.16184 | 112.98074 |

87269 | 586.5961 | 0.9645518 | 20.07087 | -147.4484 | 142.37805 |

2002 RN109 | 746.6725 | 0.9963802 | 57.99165 | -147.53176 | 170.49258 |

308933 | 780.1764 | 0.969011 | 19.46659 | 122.49628 | -162.61603 |

2013 AZ60 | 990.1281 | 0.992013 | 16.52296 | 157.9403 | -10.77058 |

Then I used the R package to produce a hierarchical cluster as follows:

1) I scaled the above table so that every column has mean 0 and variance 1.

2) I calculated the distance between any two rows (manhattan distance).

3) I submitted the scaled table to the function hclust choosing clustering method complete.

4) I used a further R function ( see rect.hclust ) to display colored rectangles at different height: the purpose is to help visualize the various clusters at different levels.

**Result**I do not know, whether these clusters have a statistical significance or not.

The left cluster maybe interesting: in fact, it consistently maintains its shape even when you cut the dendogram at a level where the second big cluster gets split into 5 subgroups.

The left cluster contains asteroid 2012 VP113 plus other 4 even more similar asteroids.

From now on I will refer to this cluster as "cluster 2" as opposed to "cluster1" made by all other KBOs without further distinctions.

A nice R function is cutree: you can tag the original table with a further column i.e. the cluster where it is supposed to belong. By doing this, you can use, for example, the function ggpairs of the GGally library to make a set of plots like these:

- in the diagonal, you can see the density distribution, each cluster being given a different color. Cluster 2 is coloured in blue.
- above the diagonal: you can see the correlation between pairs of parameters, with cluster detail and total as well.
- below the diagonal: you can see a scatter plot diagram of each pair of parameters, again the colour represents a cluster.

.... and, if you are interested in a specifc plot, you can make it alone with the ggplot function.

For example:

- let's see a scatter plot of orbital parameters a and w where we add the name of the asteroids
- finally, let's see the density distribution of orbital parameter w

__a versus w__

__w density distribution__

Kind Regards,

Alessandro Odasso

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