Astron. Astrophys. 346, 652-662 (1999)
7. Characteristics of the families
The following characteristics of the earlier mentioned families are
Besides the observation of Sect. 6.1, we further observe that
For all members of a family, the general shape of the orbits is the
same so long so the escaper remains the same.
We know that
When a body escapes, its distances from the other two bodies tend
to and F varies according to
where i and j do not
take up the values corresponding to the suffix of the escaper. This
also means that F is minimum or maximum according as the
distance between the binary is maximum or minimum. Thus we conclude
that the value of F is governed by one single close binary
approach after is attained and not
When the perturbing velocities have the same magnitude and
direction, by symmetry the relative distances between the
participating bodies remain equal. But if directions of the perturbing
velocities are not equal (keeping the magnitude of the velocities
equal), we are able to say which body amongst them is going to escape
with the formation of a binary.
The quantity (the total mass is
three), which is used as the radius of gyration by Birkhoff (1927) and
Sundman (1912) instead of moment of inertia, always hold the following
The magnitude of velocity of the latecomer (escaper) after the time
of first close approach increases and the escaper moves towards the
centre of mass.
The latecomer (escaper) must pass close to the centre of mass after
the first close approach.
In each family, the body for which the perturbing velocity is
varied never escapes.
The semimajor axis and its
eccentricity e for case (I) (Sect. 4) of the first family with
their sub- cases are shown in Figs. 4a and 4b.
Proceeding as in Szebehely (1974a), it can be shown that
for all families.
The relation is in good agreement with the numerical results. We have
shown the value of for both the sub
cases of the first family for case (I) (Sect. 4) in Figs. 5a and 5b.
It may be noted from the above Figs. 4a and 4b that for the family
1(a), when varies from
and other parameters are fixed, the
semimajor axis decreases as
increases up to a certain value and
then a rapid fall. Further, eccentricity e decreases slowly
with up to a certain value and then
there is sharp decrease and after that it increases rapidly. And for
the family 1(b) when varies from
and other parameters are kept fixed
as above, the semimajor axis and its
eccentricity e fluctuate. These trends are the same for all
other families. Further, it may also be noted in Figs. 5a and 5b that
for the first family 1(a) the escape velocity
decreases slowly as
increases up to a certain value and
after that it increases rapidly. And for the family 1(b), escape
velocity decreases as
increases up to a certain value
after that it increases rapidly.
The minimum moment of inertia and
the difference between
of the families are characterised.
For the family 1(a) for case (I) (Sect. 4),
decreases slowly up to a certain
value as increases after that it
decreases sharply and then increases rapidly where as
increases slowly up to a certain
value with after that it increases
rapidly (Fig. 6a). For the family 1(b)
fluctuate (Fig. 6b). The trends for
are the same for all other
Fig. 4a. Semimajor axis and eccentricity e Vs ; , ; ,
Fig. 4b. Semimajor axis and eccentricity e Vs ; , ; ,
Fig. 5a. Escape velocity Vs ; , ; ,
Fig. 5b. Escape velocity Vs ; , ; ,
Fig. 6a. Time to triple close approach and minimum moment of inertia Vs ; , ; ,
Fig. 6b. Time to triple close approach and minimum moment of inertia Vs ; , ; ,
© European Southern Observatory (ESO) 1999
Online publication: May 21, 1999