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Astron. Astrophys. 326, L21-L24 (1997)

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3. Formation mechanism

Our computations show that high eccentricities can be generated during multi-star gravitational interactions. The main mechanism is connected with the formation of a hierarchical triple or quadruple system. In order to identify stable configurations we use the stability criterion of Mardling and Aarseth (1997a). By using this criterion we select the stable hierarchical configuration to apply the Heggie and Rasio theory to the inner (planetary) orbit. The critical ratio of the outer periastron distance of the mass [FORMULA] to the inner apastron distance of [FORMULA] is given by

[EQUATION]

where [FORMULA] are the outer and inner eccentricities respectively, [FORMULA], [FORMULA] = 2.8, and [FORMULA] = 2. This criterion has been verified (for mass ratios in range 0.01-100 of the outer body and wide range of values for [FORMULA]) by systematic calculations (Mardling & Aarseth 1997b).

The characteristic time-scale on which a single star is captured by a mono-planetary system (hereafter MPS) is given approximately by

[EQUATION]

where [FORMULA] is the probability that a fourth star (single in this case) also lies within a given distance d, [FORMULA] is the number density of single stars, [FORMULA] is the capture cross-section, and v is the root mean square velocity of stars in the system. The cross-section for a single star to pass within a distance d of the centre of mass of a MPS is given by

[EQUATION]

where [FORMULA] is the mass of the MPS and M is the mass of the incoming star. From its point of view, the MPS is a single star-like object, and in order to form an outer binary the velocity perturbation in the encounter must be approximately the RMS velocity of the cluster stars, [FORMULA], where R is the half-mass radius of the cluster and [FORMULA] is the mean mass of the stars. Considering [FORMULA] we have

[EQUATION]

where [FORMULA] is the number of single star (without planets) and N is the total number of objects (stars+centre of mass of planetary systems). Including the stability criterion we have

[EQUATION]

where [FORMULA], and a is the MPS semi-major axis. According to the stability criterion a typical value for [FORMULA] can be about 30 (for very eccentric outer body) and using the crossing time ([FORMULA]) we have

[EQUATION]

From the calculations these systems only form in the cluster core so we must use the core parameters in Eq. (6). This equation gives [FORMULA] for the typical values of the parameters found in our calculations. The frequency of hierarchical system formation for a cluster with N =300 and 50 mono-planetary systems could be 0.01 per crossing time or about 2 during the typical cluster life-time for the range of N considered in the calculations. If the initial fraction of mono-planetary systems is larger, this process can be very important.

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© European Southern Observatory (ESO) 1997

Online publication: April 8, 1998
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