Secular evolution of asteroid orbits

(10.85) |

and

(10.86) |

as well as

(10.87) |

By analogy with the analysis in the previous section, the secular terms in the disturbing function of the asteroid, generated by the perturbing influence of the planets, cause the asteroid's osculating orbital elements to evolve in time as

where

(10.92) | ||||||

(10.93) | ||||||

(10.94) | ||||||

and | (10.95) |

However, as we have already seen, the planetary osculating elements themselves evolve in time as

(10.96) | ||||||

(10.97) | ||||||

(10.98) | ||||||

and | (10.99) |

Equations (10.88)-(10.91) can be solved to give

where

(10.104) | ||||||

(10.105) | ||||||

(10.106) | ||||||

and | (10.107) |

as well as

(10.108) | ||||||

and | (10.109) |

The parameters, and appearing in Equations (10.100)-(10.103) are the eccentricity and inclination, respectively, that the asteroid orbit would possess were it not for the perturbing influence of the planets. These parameters are usually called the

Figure 10.6 shows the osculating eccentricity plotted against the sine of the osculating inclination
for the orbits of the first 100,000 numbered asteroids (asteroids are numbered in order of their discovery).
No particular patten is apparent. Figures 10.7 and 10.8 show the free eccentricity plotted against the sine of the
free inclination for the same 100,000 orbits. In Figure 10.7, the free orbital elements
are determined from standard Laplace-Lagrange secular evolution theory, whereas in Figure 10.8 they
are determined from Brouwer and van Woerkom's refinement of this theory. It can be seen that many of the points representing the
asteroid orbits have condensed into clumps. These clumps, which are somewhat clearer in Figure 10.8 than in Figure 10.7, are known as *Hirayama families* after their discoverer, the Japanese astronomer Kiyotsugu Hirayama (1874-1943). It is thought that the asteroids making
up a given family had a common origin; most likely due to the break up of some much larger body (Bertotti et al. 2003).
As a consequence of this origin, the asteroids originally had similar orbital elements. However, as time
progressed, these elements were jumbled by the perturbing influence of the planets. Thus, only when this
influence is removed does the commonality of the orbits becomes apparent. Hirayama families are named
after their largest member. The most prominent families are the (4) Vesta, (15) Eunomia, (24) Themis, (44) Nysa, (158) Koronis, (221) Eos, and (1272) Gefion families. (The number in brackets is that of the corresponding asteroid.)