4 A dynamical ejection in Orion

4.1 AE Aurigae & $\mu$ Columbae

Blaauw & Morgan (1954) drew attention to the isolated stars AE Aur (O9.5V) and $\mu$ Col (O9.5V/B0V), which move away from the Orion star-forming region (e.g., McCaughrean & Burkert 2000) in almost opposite directions with comparable space velocities of $\sim$100 km s^-1 (Fig. 11, stars 5 and 6 in Table 3). Blaauw & Morgan suggested that "... the stars were formed in the same physical process 2.6 million years ago and that this took place in the neighborhood of the Orion Nebula.'' The past orbits of AE Aur and $\mu$ Col intersect on the sky near the location of the massive highly-eccentric double-lined spectroscopic binary $\iota$ Ori (O9III+B1III, see Stickland et al. 1987). This led Gies & Bolton (1986) to suggest that the two runaways resulted from a dynamical interaction also involving $\iota$ Ori: "... $\iota$ Ori is the surviving binary of a binary-binary collision that ejected both AE Aur and $\mu$ Col.''

Figure 7: Contours of minimum separation for the pairs AE Aur-$\mu$ Col ( top), AE Aur-$\iota$ Ori ( middle), and $\mu$ Col-$\iota$ Ori ( bottom). The contours are spaced every 1 pc with the outermost contour being 10 pc. The ordinates represent straight lines in the distance-distance plane, defined as indicated in the top right of each panel

  
4.2 Data

Table 3 lists the data for AE Aur, $\mu$ Col, and $\iota$ Ori. We adopt Stickland's et al. (1987) radial velocity for $\iota$ Ori ( $\vec{v}_\gamma$). For the radial-velocity errors for AE Aur and $\mu$ Col we use the largest errors quoted in either the Catalogue de Vitesses Radiales Moyennes Stellaires (Barbier-Brossat 1989), the Hipparcos Input Catalogue (Turon et al. 1992), the Wilson-Evans-Batten Catalogue (Duflot et al. 1995), or in the SIMBAD database.

  
4.3 Simulations

To investigate the hypothesis that the three stellar systems, AE Aur, $\mu$ Col, and $\iota$ Ori, were involved in a binary-binary encounter, we retrace their orbits back in time to find the minimum separation between them. As in Sect. 3, we explore the parameter space determined by the errors of, and correlations between, the observables.

Even with the unprecedented accuracy in trigonometric parallaxes obtained by the Hipparcos satellite, the errors on the individual distances are rather large: $D_{\rm AE~Aur} = 446_{-111}^{+220}$ pc, $D_{\mu\ {\rm Col}} = 397_{-\phantom{1}71}^{+110}$ pc, $D_{\iota\ {\rm Ori}} = 406_{-\phantom{1}96}^{+185}$ pc. We therefore first determine which distances are most likely to agree with our hypothesis, and then study the effect of the measurement errors on the other observables. For each pair of stars, Fig. 7 shows contours of minimum separation between the respective orbits as a function of their present distances. The distances of the stars for which the orbits have a small minimum separation are strongly correlated, i.e., if the distance of star i increases that of star j also needs to increase to obtain a small minimum separation. We therefore choose to show the contours of constant minimum separation with respect to this correlation. The vertical axes thus show offsets from the straight line in the distance vs. distance plane defined by the equation in the top right of each panel.

We start each simulation with a set of positions and velocities which are in agreement with the observed parameters and their covariance matrices (<3$\sigma$). Furthermore, we require the distances of the stars to fall within the 10 pc minimum-separation contours of Fig. 7. We then calculate the orbits of AE Aur, $\mu$ Col, and $\iota$ Ori. We define the separation between the three stellar systems, $D_{\rm min}(\tau)$, as the maximum deviation of the objects from their average position, i.e., $D_{\rm min}(\tau) = {\rm max}\vert\vec{x}_{j} - \bar{\vec{x}}\vert$ for j = AE Aur, $\mu$ Col, and $\iota$ Ori, where $\bar{\vec{x}} = \frac{1}{3} (\vec{x}_{{\rm AE\ Aur}} + \vec{x}_{\mu\ {\rm Col}} + \vec{x}_{\iota\ {\rm Ori}})$ is the mean position and $\vec{x}_j$the position of star j. The time $\tau_0$ at which $D_{\rm min}(\tau)$ reaches a minimum is considered to be the time of the encounter, i.e., the kinematic age.

Figure 8: Left: Distribution of minimum separations between AE Aur, $\mu$ Col, and $\iota$ Ori, $D_{\rm min}(\tau_0)$, of $10\,000$ Monte Carlo simulations of the stellar encounter. Right: The $D_{\rm min}(\tau_0)$ distribution for three randomly drawn points from three spherical Gaussians with standard deviations of $\sigma = 4$ pc (solid line and shaded), $\sigma = 2$ pc (dotted line), and $\sigma = 6$ pc (dashed line). The $D_{\rm min}(\tau_0)$distribution for $\sigma = 4$ pc is a good representation of the distribution in the left panel. Four pc is the typical spread in the end positions of the orbits due to the errors on the present day velocity ($\sim$2 km s-1). The dotted and dashed histograms have been normalized such that their shapes can be compared with the solid histogram

We computed 2.5 million orbits, of which 114 yielded $D_{\rm min}(\tau_0) < 1$ pc with $\tau_0 =$ 2-3 Myr. One of the simulations resulted in a minimum separation of 0.019 pc which is equal to $4\,000$ AU (Fig. 8). The small number of simulations with small minimum separations is due to (i) the large number of parameters involved (i.e., 18) and (ii) the three-dimensional nature of the problem (cf. Sect. 3.3).

We have numerically determined the distribution of the minimum separations $D_{\rm min}$ of three points drawn from a three-dimensional Gaussian error distribution (the analytic results of Appendix A are valid only for two Gaussians). We randomly draw three points from three spherical three-dimensional Gaussians (with standard deviation $\sigma$) and determine $D_{\rm min}$. The Gaussians have the same mean positions. The resulting distribution for $\sigma = 4$ pc resembles the real one remarkably well (Fig. 8). A distance uncertainty of four pc is consistent with the $\sim$2 km s^-1 uncertainties in the velocities of the runaways and $\iota$ Ori: 2 km s^-1 over $\sim$2 Myr results in a displacement of $\sim$4 pc. Thus, the data and their errors are consistent with the hypothesis that $\sim$2.5 Myr ago AE Aur, $\mu$ Col, and $\iota$ Ori were in the same small region of space.

Figure 9: Properties of the parent cluster of the runaways AE Aur and $\mu$ Col and the binary $\iota$ Ori obtained from our Monte Carlo simulations. First row: cluster properties plotted vs. the cluster distance. The grey dots denote the median values of the cluster properties for distance-bins of 25 pc. Second row: histograms of the predicted cluster properties. The tick marks on the vertical axis have a spacing of 1000. Third row: histograms of the predicted cluster properties when the mass of $\mu$ Col is changed by $-1~M_\odot$. Note that the distance, time, and radial-velocity histograms do not change significantly. The tick-mark spacing along the vertical axis is similar to that in the second row

4.4 Interpretation

The nominal observed properties of the runaway stars AE Aur and $\mu$ Col and the binary $\iota$ Ori are consistent with a common origin $\sim$2.5 Myr ago. The most likely mechanism that created the large velocities of the runaways and the high eccentricity of the $\iota$ Ori binary is a binary-binary encounter, as suggested by Gies & Bolton (1986). The normal rotational velocities of both runaways (25 km s^-1 and 111 km s^-1) and the normal helium abundance of AE Aur (Table 3, see Blaauw 1993, Fig. 6) also suggest that these runaways were formed by the dynamical ejection scenario. The helium abundance of $\mu$ Col is unknown.

  
4.5 Parent cluster

To find the cluster, or region of space, where the encounter between AE Aur, $\mu$ Col, and $\iota$ Ori took place we assume that the center of mass velocity of the three objects is identical to the mean velocity $\vec{v}_{\rm clus}$ of the parent cluster. Then

$$\vec{v}_{\rm clus} = \frac{\sum_j M_j \vec{v}_j}{\sum_j M_j},$$ (2)

for j = AE Aur, $\mu$ Col, and $\iota$ Ori. For each star we estimate the mass by interpolating the mass vs. spectral-type calibration of Schmidt-Kaler (1982, Table 23). We obtain $15.9~M_\odot$ for AE Aur and $\mu$ Col, and $22.9~M_\odot$ and $14.9~M_\odot$ for the primary and secondary of $\iota$ Ori, respectively ( $37.8~M_\odot$ for the binary system). We use the cluster velocity ( $\vec{v}_{\rm clus}$) and the mean position of the three stellar systems at the moment of the encounter to integrate the orbit of the ensemble of stars $\tau_0$ Myr into the future. The position and velocity at the end of this integration should coincide with the present-day properties of the parent cluster. We extend the Monte Carlo simulations described in Sect. 4.3 to include the integration of the orbit of the "cluster'' forward in time. Figures 9 and 10 summarize the results. Panel b of Fig. 10 shows that the distances of the three stars and the predicted cluster distance are tightly correlated (see also Fig. 7); all distances increase when the cluster distance increases. A consequence of this tight correlation is that as soon as the distance to one of the objects is known, all other distances are fixed.

Figure 10: Properties of the parent cluster of the runaways AE Aur and $\mu$ Col and the binary $\iota$ Ori. a) Proper motions and their errors (grey circles) for all stars in the Tycho 2 Catalogue (filled triangles) within an area of 0 $.\!\!^\circ$4 by 0 $.\!\!^\circ$4 centred on the Trapezium cluster. The large grey dot denotes the average of the predicted cluster proper motion for the Monte Carlo simulations. b) Distances of the runaway stars as a function of the cluster distance in the Monte Carlo simulations. The different grey scales are labeled in the panel. The filled circles and their error bars denote the observed distances of the stars derived from the Hipparcos parallaxes (prlx) and the open circles denote the distances derived from photometry (phot) (Gies 1987). c) the biases on the predicted cluster distance as discussed in Sect. 4.5.1. The filled and open symbols denote the mean and median, respectively, of the cluster distance distributions. The circles include only the first bias, the aiming effect, and the triangles include both the aiming effect and the ``incorrect'' Hipparcos parallax of $\mu$ Col. For clarity, the circles and triangles are displaced -3 and 3 pc, respectively. The dotted line indicates the mean cluster distance based on the Monte Carlo simulation

  
4.5.1 Biases and measurement errors

Two effects influence the mean cluster properties as predicted by the Monte Carlo simulations. First, it is easier to hit a target from close by than from far away, i.e., a larger range of velocities (within the errors) is consistent with the encounter hypothesis when the distance between the star and the encounter point is small (the "aiming effect''). We simulate this effect in the following way. We assume a range of cluster distances, 350-500 pc. For each distance we use Fig. 9 (the gray dots in the first row) to determine the other phase-space coordinates of the parent (position on the sky, proper motion, and radial velocity). With these "observables'' we calculate the three-dimensional velocity of the cluster, corrected for Solar motion, and determine its position at a time $\tau_0$ (see first row in Fig. 9) in the past. We neglect the variation of the Galactic potential, ignore Galactic rotation, and use the linear velocity, to speed up the calculations. This past position of the cluster combined with the present three-dimensional positions of AE Aur, $\mu$ Col, and $\iota$ Ori (based on the present positions on the sky and the distances from Fig. 10 panel b) gives the velocities of the three stellar systems today, using $\tau_0$ as the time difference. These "observed'' properties are then used as input for the Monte Carlo simulations described above to investigate the influence of the aiming effect on the predicted cluster distance. The circles in Fig. 10 panel c display the bias in the cluster distance.

Secondly, the trigonometric distance of $\mu$ Col, $D_{\mu\ {\rm Col}} = 397_{-\phantom{1}71}^{+110}$ pc, is smaller (2$\sigma$) than the observed photometric distance, $\sim$750 pc (e.g., Gies 1987). The photometric distance is reliable since $\mu$ Col is located in a region free of interstellar absorption. This difference between the trigonometric distance and the "real'' distance results in an additional bias towards smaller distances for the stars and the cluster. In our Monte Carlo simulation we draw the parallaxes, like all other observables, from a Gaussian centred on the observed value and with a width equal to the observed error. For the Hipparcos distance of $\mu$ Col this means that less than $\sim$10% of the random realizations will be consistent with the photometric distance. And because the distances of the three stellar systems and the cluster are correlated (see Fig. 10 panel b), the other stars also need to be at smaller distances for the encounter to take place. This effect will result in a mean cluster distance (the mean of the Monte Carlo simulations) which is underestimated. We simulated this effect in a similar manner as the aiming effect. The results on the mean cluster distance in the Monte Carlo simulations, aiming effect and the parallax of $\mu$ Col, are shown as the triangles in Fig. 10 panel c.

4.5.2 Cluster properties and identification

Taking the biases on the cluster distance into account, we reconstruct the present-day properties of the parent cluster of the stars AE Aur, $\mu$ Col, and $\iota$ Ori. The mean cluster distance from our Monte Carlo simulations is 339 pc (right most panel in the second row of Fig. 9). The cluster distance corrected for biases is 425-450 pc. Using this distance we determine the other properties of the cluster (first row of Fig. 9), and summarize them in Table 4. Figure 9 (right most panel of the first row) indicates that the encounter happened 2.5 Myr ago; this obviously is a lower limit to the age of the cluster. Figure 11 shows the region of the sky where the parent cluster should be located: the Orion Nebula. The black contours in the bottom panel show the distribution of $(\ell ,b)$ of the parent cluster obtained from the Monte Carlo simulations.

   

Table 4: Predicted properties of the parent cluster of the stellar systems AE Aur, $\mu$ Col, and $\iota$ Ori: the cluster distance $D_{\rm parent}$, the sky position in equatorial $(\alpha ,\delta )$and Galactic $(\ell ,b)$ coordinates, the proper motions $(\mu _{\alpha \ast },\mu _\delta )$ and $(\mu _{\ell \ast },\mu _b)$, and the radial velocity $v_{\rm rad}$. The predicted distances of the runaways and $\iota$ Ori if there was an encounter: $D_{\rm AE~Aur} = 430$ pc, $D_{\mu~{\rm Col}} = 600$ pc, and $D_{\iota~{\rm Ori}} = 440$ pc

			$M_{\mu~{\rm Col}} - 1~M_\odot$
$D_{\rm parent}$	425-450		425-450	pc
$(\alpha ,\delta )$	(84 $.\!\!^\circ$0,-5 $.\!\!^\circ$8)		(83 $.\!\!^\circ$9,-5 $.\!\!^\circ$2)
$(\mu _{\alpha \ast },\mu _\delta )$	(1.7,-0.8)		(1.7,-0.2)	mas yr-1
$(\ell ,b)$	(209 $.\!\!^\circ$4,-19 $.\!\!^\circ$4)		(208 $.\!\!^\circ$9,-19 $.\!\!^\circ$2)
$(\mu _{\ell \ast },\mu _b)$	(1.3,1.2)		(0.9,1.4)	mas yr-1
$v_{\rm rad}$	28.3		27.6	km s-1

Of all clusters in this active star-forming region the Trapezium cluster (NGC 1976) is the most likely parent cluster for the following reasons.

1:

The cluster is young. Palla & Stahler (1999) find a mean age of $\sim$2 Myr based on theoretical pre-main-sequence tracks, and established that the first stars formed not more than 5 Myr ago. Thus, the Trapezium is old enough to have produced the runaways;

2:

The Trapezium is one of the most massive, dense clusters in the Solar neighbourhood. Estimates for the stellar density are $>20\,000$ stars pc^-3 for the inner 0.1-0.3 pc (e.g., McCaughrean & Stauffer 1994; Hillenbrand & Hartmann 1998). These high stellar densities favor dynamical interactions within the cluster core;

3:

The Trapezium shows a strong mass segregation (Zinnecker et al. 1993; Hillenbrand & Hartmann 1998). Five of the six stars more massive than $10\, M_\odot$ are in the centre. This concentration of massive stars increases the probability for dynamical interactions between these stars;

4:

The binary fraction in the Trapezium cluster is at least as high as that of the Solar-type field stars, i.e., $\sim$60% (Prosser et al. 1994; Petr et al. 1998; Simon et al. 1999; Weigelt et al. 1999). This means that enough binary systems are available for binary-binary or binary-single-star interactions to become efficient in expelling stars from the cluster.

The mean astrometric properties and the radial velocity of the Trapezium agree perfectly with those predicted by our Monte Carlo simulation. The distance to the Trapezium is estimated to be 450-500 pc (Walker 1969; Warren & Hesser 1977a, 1977b, 1978; Genzel & Stutzki 1989); we predict 425-450 pc. The observed radial velocity of the Trapezium is 23-25 km s^-1 (Johnson 1965; Warren & Hesser 1977a, 1977b; Abt et al. 1991; Morrell & Levato 1991); we predict $\sim$28 km s^-1. The absolute proper motion of the Trapezium is ill-determined, but is known to be small (e.g., de Zeeuw et al. 1999). We collected all stars, within a 0 $.\!\!^\circ$4 by 0 $.\!\!^\circ$4 region centred on the Trapezium, based on the Tycho 2 Catalogue (Høg 2000), and plot the proper motions in Fig. 10a. The proper motions agree with the predicted cluster proper motion.

Table 4 shows that the predicted position on the sky of the parent cluster does not fully agree with the position of the Trapezium (see Fig. 11). Here it is important to remember that we did not allow for any errors on the stellar masses used in Eq. (2). We investigate the effect of mass errors by changing the masses and running a new set of Monte Carlo simulations. We find that (i) the results are insensitive to the mass of $\iota$ Ori: a change as large as $\pm$5 $M_\odot$ produces no noticeable change in the cluster properties, and (ii) the sky position of the parent cluster and its proper motion depend on the mass ratio of AE Aur and $\mu$ Col. Changing the mass of $\mu$ Col by $-1~M_\odot$ or the mass of AE Aur by $+1~M_\odot$ shifts the predicted sky position of the parent cluster to that of the Trapezium cluster (Fig. 11). A mass change in the other direction, $+1~M_\odot$ for $\mu$ Col and $-1~M_\odot$ for AE Aur, creates a similar shift in the opposite direction. There are indications from spectral-type determinations that $\mu$ Col is indeed slightly less massive than AE Aur. Most spectral-type determinations of $\mu$ Col give O9.5V; however, Blaauw & Morgan (1954) and Paper I quote B0V and Houk (1982) quotes B1IV/V.

Figure 11: Top & middle: Orbits, calculated back in time, of the runaways AE Aur (dotted line) and $\mu$ Col (solid line) and the binary $\iota$ Ori for one of the Monte Carlo simulations described in the text. The top panel shows the distance vs. Galactic longitude of the stars. The middle panel shows the orbits projected on the sky in Galactic coordinates. The starred symbols depict the present position of the three stars. The stars met $\sim$2.5 Myr ago. Using conservation of linear momentum, the orbit of the parent cluster (grey solid line, see blow up) is calculated from the time of the assumed encounter to the present. The large circles denote all stars in the Hipparcos Catalogue brighter than V = 3.5 mag; filled circles denote O and B stars, open circles denote stars of other spectral type. The small circles denote the O and B type stars with $3.5~{\rm mag} \le V \le 5$ mag (cf. Fig. 1 in Blaauw & Morgan 1954). The Orion constellation is indicated for reference. Bottom: The predicted position of the parent cluster (black contours) together with all stars in the Tycho Catalogue (ESA 1997) in the field down to V = 12.4 mag. The size of the symbols scales with magnitude; the brightest star is $\iota$ Ori. The Trapezium and $\iota$ Ori are indicated. The black and dark grey lines are the past orbits of $\iota$ Ori and the Trapezium, respectively (see top panel). The triangle denotes the predicted present-day position of the parent cluster for this particular simulation. The grey contours display the IRAS 100 micron flux map, and mainly outline the Orion Nebula

We note that the calibration of Vanbeveren et al. (1998) gives a mass of 38.6 $M_\odot$ for $\iota$ Ori, similar to that found with the Schmidt-Kaler calibration, but increases the masses of AE Aur and $\mu$ Col to 21.1 $M_\odot$. This does not change our results, as it is the ratio of the runaway masses that determines the predicted current position of the parent cluster.

In summary, the position, distance, proper motion, and radial velocity of the Trapezium cluster fall within the range predicted by our Monte Carlo simulations. Furthermore, the youth, extreme stellar density, mass segregation, and the high binary fraction make it the best candidate for the parent cluster of the runaways AE Aur and $\mu$ Col and the binary $\iota$ Ori. Finally, it is the only likely candidate in this region of the sky.