KINEMATICS OF THE ORION TRAPEZIUM BASED ON DIFFRACTO-ASTROMETRY AND HISTORICAL DATA

There exist a number of systematic errors associated with the WFPC2 measurements which have been extensively analyzed in the past. The most important are: geometric distortion (Kozhurina-Platais et al. 2003; Anderson & King 2003), the 34th row error (Anderson & King 1999), charge transfer efficiency (CTE; Holtzman et al. 1995; Stetson 1998; Whitmore et al. 1999; Dolphin 2000), and plate scale variations (McMaster & Biretta 2008; Gonzaga & Biretta 2010).

The OT images used in this analysis were corrected for geometric distortion and the 34th row error using the formulae given by the authors listed above. In the case of geometric distortion, there are published transformation coefficients only for filters F300W, F555W, and F814W. For those images taken with other filters, the distortion coefficients used were those pertaining to the nearest filter in wavelength, for which such coefficients exist in the literature.

It is known that CTE affects astrometry in Advanced Camera for Surveys Wide Field Channel (Kozhurina-Platais et al. 2007). However, to our knowledge, there is no correction for CTE effects on WFPC2 astrometry (Gonzaga & Biretta 2010). In addition, according to McMaster & Biretta (2008, p. 103), "Images with high background signal suffer relatively little charge loss because the background signal fills the traps and prevents them from robbing the charge packets during readout." Since the OT region is projected over the Orion Nebula, the background that these images have is relatively high (∼200 e⁻ for images with exposure times ⩾10 s), rendering the CTE effect negligible even for the fainter E and F component spikes.

In order to account for possible changes in plate scale due to the use of different wavelengths, we have used all the plate scales reported in Table 5.10 of McMaster & Biretta (2008), and obtained interpolated values for other filters.

Separations and P.A.s between OT stellar pairs were obtained from the WDS. Data in this compilation are given with no uncertainties, so we selected only those for which we can assign realistic errors. For this reason, we only used data for those observers classified as "Best" and "Good" in Table 1 of Allen et al. (2004). As stated by these authors, the error bars in separation were assigned by C. Worley "based on a life-time of experience." In that study, only the measures for the relative separations were used, because the historical P.A.s were considered less reliable.

In the present study, the uncertainties for the historical P.A. observations were estimated from the dispersions of different measurements by the same observer, pair of components, and epoch. These measurements were found to be considerably less reliable than those for the separations.

The HST public database (Mikulski Archive for Space Telescopes ² ) was meticulously inspected in order to find OT images with the highest astrometric quality. The selection criteria were the following: (1) We only use images taken with the WFPC2. Since the WFPC2 is fixed on the axis of the HST, the images taken with it have a very good optical quality. (2) The OT image must be present in one of the four detectors, to avoid inter-chip errors due to different reference frames. (3) The stellar components to be measured must not lie within the first ∼50 x and y pixels; in this region the pyramid mirror makes it impossible to perform accurate astrometry. (4) The background of the images must be ∼200 e⁻ in order to avoid position shifts due to CTE losses. (5) The diffraction pattern produced by the supports of the secondary mirrors (of both telescope and camera) must be present in at least two images of the OT components.

We used 44 images that fulfilled these criteria. These images were downloaded in waivered FITS (c0f.fits) format and were already calibrated by the standard pipeline procedure. Their characteristics are listed in Table 1.

Images are distributed in eight epochs, spanning from 1995 March 21 to 2007 November 6. 56% of the images were taken in 1995, 16% in 2007, and the rest during different years of the observing period (1995–2007). The images have a mean exposure time of ∼120 s. 68% of them were taken using narrow-band (N) filters, while 28% were taken using wide-band (W) and medium-band (M) filters. 36% were taken with Detector 4, 27% with Detector 3, and the rest with Detector 1; no images taken with Detector 2 were used in this analysis. As an example, Figure 1 shows one of the images used in this work.

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Ruelas-Mayorga et al. (2011) presented an analysis of the accuracy with which the DA technique is able to recover the photocenter of simulated point-spread functions of the WFPC2. We present here the results of a similar analysis of the DA precision on real WFPC2 images. In order to do this, we selected 25 images, from our data set, taken in one epoch only: 1995 March 21; in this way we avoid errors associated with individual motions of the stars. These images were taken with Detectors 1 and 4 (11 and 14 images, respectively) in mid- and narrow-band filters (3 and 22 images, respectively).

In every one of the WFPC2 images mentioned in the previous paragraph, the stellar photocenter positions, as well as their precision, were measured as indicated in Section 2 of Ruelas-Mayorga et al. (2011). The precision of all these measurements was analyzed as a function of detector, filter, spike's effective signal-to-noise ratio (eS/N), and average effective length; the last two quantities are defined below.

To determine the eS/N of one single point in the spike, a cut—21 pixels for Detectors 2, 3, and 4 (WF) and 33 pixels for Detector 1 (PC)—perpendicular to the spike at that point is performed. The point's eS/N is obtained as follows: (1) the length of the cut is divided in three equal sections; (2) the average intensity of the two outer sections is calculated; (3) the maximum of the central section is obtained; and (4) the eS/N is calculated as the ratio of the maximum value calculated in (3) to the average value derived in (2). The spike's eS/N was computed as the average of the eS/N values of all the points along the spikes used for establishing a stellar position.

The average effective length of a spike associated with a stellar photocenter measurement is defined as the average number of points along the four spikes used to determine the photocenter position.

As an example of the precision determination we present, in Figure 2, a plot of the distribution of uncertainties as a function of spike average length. It is clear from this figure that the value of the precision improves, as expected, with the average length of the spikes. It seems that the precision may not be improved further than ∼0.01 pixels with this technique. However, it was found that the DA precision is, for all cases, smaller than 0.11 pixels.

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In each of our 44 images, we measured all present OT components together with at least 9 (29 at most) unsaturated field stars. The position of the saturated stars was measured using the DA technique, while the position of the unsaturated ones was estimated using Lorentzian fitting algorithms. Positions were grouped by epoch and a reference frame was established for each one of these; we used as a reference frame the image with the largest number of measured unsaturated positions. In Table 2 we list the epochs of all the reference frames, together with their corresponding filter, detector, plate scale, and number of matched frames. Using the geomap task of IRAF ³ and all unsaturated positions of the epoch reference frame as pivots, we obtain a transformation between each image and its epoch reference frame. These transformations allow us to map the positions of the OT components (A–F) on this epoch pixel-corrected reference frame. These positions are reported in Table 3.

Using the positions of the photocenters reported in Table 3, together with their corresponding plate scales in Table 2, we were able to measure, for every pair of components, the separation (in arcseconds) and the P.A. (in degrees, measured by convention, from north to east).

The north direction was established using the group keyword ORIENTAT given in the header of each WFPC2 image. This keyword records the angle, in degrees, between the north and the columns of the detector; it is measured from north to east. Determination of the north direction is limited by the HST guide star positions, the mode in which these stars were acquired by the Fine Guidance Sensors (FGS), and the transformation between FGS and WFPC2. In all of our images, the FGS's acquisition mode was FINE; this mode renders a typical rms pointing precision better than 2–5 mas (Fruchter & Sosey 2009). These figures, transformed to the WFPC2, provide an absolute astrometry accuracy between 0.5 and 1 arcsec (Gonzaga & Biretta 2010).

The entire list of measurements, from both WDS and DA, can be obtained from the authors by request.

A combination of OT-astrometric data from the WDS and DA measurements of 44 OT HST images allows us to present in the next section the internal motions of the OT.

To determine the internal kinematics of the OT components, we performed linear least-squares (LSQ) fits to the separation and the P.A., for both WDS and DA data, as a function of time.

As examples of the relative motions of the OT components, we present Figures 3–6, in which we plot the values for the separation and the P.A. of the component pairs AC, AE, CB, and CD, as a function of time. The straight line resulted from linear LSQ fits to the points.

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Since the precision of the P.A. historical measurements is low, as stated in Section 3.1, and the time base of the DA measures is relatively short, the rates of change derived from the fit to the P.A. data are significantly less reliable than those obtained for the fits to the separation data.

Tests for membership of OT components A–E to the Trapezium were already performed by Allen et al. (2004). However, in that study it was not possible to establish the physical membership of component F to the OT, due to the fact that component C shows widely discrepant values for its proper motion (Jeffers et al. 1963; Mason et al. 2001) and hence the proper-motion test for pair CF could not be carried out. Given the large transverse velocity of F relative to C derived from our measures for the P.A. and separation velocities (11.2 ± 0.8 km s⁻¹ and 2.7 ± 0.9 km s⁻¹, respectively), we think it is a foreground star. It is probably related to the stellar population located in front of the ONC, as proposed by Alves & Bouy (2012). In view of this fact, we decided to disregard the kinematical information of this component.

The relative motions of components A–E of the OT are summarized in Table 4. In this table, the first column indicates the pair of OT components, the second and third the rate of change in P.A. and separation, respectively. The figures in the last two columns were transformed to km s⁻¹ adopting a distance to the Orion Nebula of 414 ± 7 pc (Menten et al. 2007). The precision of these relative velocity determinations was found to be ∼1.0 km s⁻¹; this value corresponds to the average of uncertainties in Table 4.

Finally, we note that the results presented in this section are fairly consistent with those published by Allen et al. (2004). These authors determine the magnitude of the relative motions between pairs AB, AC, and AE as +0.18, −0.11, and +0.36 arcsec per 100 yr, respectively, disregarding the changes in P.A. A transformation of these motions to transverse velocities results in +3.6, −2.2, and +7.1 km s⁻¹, respectively, again using 414 ± 7 pc for the distance to the Orion Nebula (Menten et al. 2007).

θ¹ Ori A is a spectroscopic and eclipsing binary (V1016 Ori). Its systemic radial velocity has been obtained by Vitrichenko et al. (1998) and Stickland & Lloyd (2000). Based on essentially the same previously published data and a few additional data points of their own, both groups obtain the value given in Table 5. The star is, in addition, an interferometric binary under close scrutiny (see Preibisch et al. 1999; Close et al. 2012).

Itself a spectroscopic and eclipsing binary, the orbital parameters of θ¹ Ori B (BM Ori) have been obtained by several authors. Vitrichenko & Klochkova (2004) notice that the values of the systemic velocity published for this system differ significantly between authors and, hence, between epochs. To reconcile previous results with their own, these authors postulate a third component in a 3.57 yr, highly eccentric orbit. In Table 5 we have adopted the systemic velocity of the putative triple system, as given by Vitrichenko & Klochkova (2004). This system is part of a "mini-cluster"; see Close et al. (2012).

θ¹ Ori C, the brightest member of the Trapezium, is an interferometric binary whose primary component is probably a spectroscopic binary system (Vitrichenko 2002a). Complicated by line-profile periodical variability associated with the oblique magnetic rotator character of the brightest component, no reliable systemic radial velocity is available for this binary. We have adopted 23.6 km s⁻¹, a value obtained by Kraus et al. (2009) by adjusting published radial velocity values for the primary component to that derived from the interferometric binary orbit. No error was associated to this barycentric velocity in that paper, but its value is nearly equal to the average of those obtained recently by Lehmann et al. (2010) and Vitrichenko et al. (2012) for the close binary (P ≃ 61 days). A conservative guess of the error associated with it is ±2 km s⁻¹.

It is frequently assumed that θ¹ Ori D is a single star. However, since the time that its radial velocity was first measured (Plaskett & Pearce 1931), it was considered to be a spectroscopic binary. Indeed, Vitrichenko (2002b), using all available velocity data, including his own measurements on IUE spectra, derived preliminary orbital elements, with two possible solutions for the orbital period that differ from each other by a factor of two. The value for the systemic velocity of this object given in Table 5 is that for the shorter period solution, preferred by the author, but the error we associate with it is three times larger than that given in his paper, since it is representative of the difference between the systemic velocity as derived from both solutions. In addition, this star is probably a visual binary (Close et al. 2012).

The values for the systemic velocity for θ¹ Ori E and F come, respectively, from Costero et al. (2008) and R. Costero et al. (in preparation).

We stress the fact that the uncertainties in Table 5 are probably gross underestimates. The radial velocities quoted are also very uncertain.

As stated above, several of the OT components are multiple stars. In principle, the orbital motion of a secondary component around its primary may produce displacements of the primary that could be confused with relative motions with respect to other components of the OT. However, we note that the only component whose orbital motion could be of the order of the DA errors in the measures is component C. The maximum relative orbital displacement that component C could show (see Table 3 in Kraus et al. 2009) is 43 mas over one orbital period. Since the mass ratio of the components is at most 0.23 (Kraus et al. 2009), the displacement due to orbital motion of component C would be at most 10 mas, smaller than our measurement errors, which are of the order of 20 mas for the relative position. All other components have either spectroscopic companions or else very large separations with correspondingly long periods (much larger than the span of our measurements).

Particularly, in the case of component A, the system A1–A2 is probably a physical binary (Close et al. 2012). Assuming a circular orbit for the system and 0° inclination, a period of about 400 yr is deduced from the change in P.A. of 092 ± 007 yr⁻¹ (Close et al. 2012). In this case, taking a separation of 020 and a mass ratio of 0.2 (Weigelt et al. 1999), component A1 will show an oscillation, about the barycenter, of 008 in 200 yr. Such an oscillation is much smaller than the precision of the WDS data, and hence undetectable.

On the other hand, the WFPC2 measurements are essentially coeval with those of Close et al. (2012), who report a separation rate of −1.4 ± 0.22 mas yr⁻¹ between A1 and A2; this translates into a 0.28 mas yr⁻¹ movement of A1 toward the barycenter, which is an order of magnitude smaller than our reported 2.8 ± 0.6 mas yr⁻¹ rate of separation between A and E.

The effect of the orbital motion of A1–A2 on the P.A. between A and E on the WFPC2 measurements depends on the configuration between A1, A2, and E, which, during the observation period, was such that the three stars were nearly aligned; consequently, the P.A. yearly rate of change reported by Close et al. (2012) for A1–A2 (092 yr⁻¹) translates into almost 01 change in P.A. for pair AE during the 12 yr lapse of the WFPC2 measurements used. In nearly 200 yr, the change in the P.A. of the pair of objects A1–E would oscillate, assuming a circular apparent orbit for A1–A2, with an amplitude of 09. These effects are of the same order as the errors of a single WFPC2 and WDS datum, respectively, and consequently undistinguishable, as can be noted in Figure 4.

We conclude that the orbital motions due to multiplicity of the OT members do not affect either the measurements carried out by the DA technique or those of the WDS compilation.

The most important kinematical results are as follows:

• 1.  
  The value we derive for the rate of change of the separation of component E relative to A, 5.5 ± 1.2 km s⁻¹, although smaller than that reported by Allen et al. (2004; 7.1 km s⁻¹ after correcting it for a distance of 414 pc to the ONC) and Sánchez et al. (2008; 6.9 ± 1 km s⁻¹), is still large enough for this component to be escaping from the Trapezium. Indeed, the escape velocity of the OT system is ∼6 km s⁻¹ according to Allen et al. (1974). The space velocity of component E relative to A (the vector sum of 5.5 ± 1.2 km s⁻¹ for the separation velocity, and 6.3 ± 2.0 km s⁻¹ for the relative radial velocity; see Table 5) is at least 8.3 ± 2.3 km s⁻¹, which is larger than the escape velocity of the OT.
• 2.  
  Although component F also appears to be escaping from the system, we were not able to confirm its physical membership to the OT. It is probably a foreground star, physically unrelated to the Trapezium. Using the photometric data of Da Rio et al. (2009) and Muench et al. (2002), we find that the visual total extinction (A_v ) of star F is at least 0.4 mag smaller than that of the bright Trapezium components, adopting the spectral type of component F (B8) given by Herbig (1950). This result places F in front of the OT, probably in the foreground veil (O'Dell 2001), and is consistent with its belonging to the abundant stellar population in front of the Trapezium Cluster proposed by Alves & Bouy (2012).
• 3.  
  The velocity dispersion we find for components A, B, C, and D is 1.1 km s⁻¹ in one coordinate. Component E is escaping, and component F is probably a foreground star; hence, they have not been considered in this estimate. Of course, a velocity dispersion based on only four stars is of doubtful significance, but it is consistent with the OT being a bound system. Even if we include component E, the velocity dispersion in one coordinate increases only to 1.8 km s⁻¹. The value we estimate is comparable to that found by van Altena et al. (1988) (1.37 km s⁻¹, when adjusted to a distance of 414 pc) for a much larger sample of member stars of the ONC. This velocity dispersion is significantly smaller than that found by Jones & Walker (1988) for a larger sample of ONC stars, but, as these authors remark, if they restrict their sample to the brighter stars, those with magnitudes comparable to the van Altena et al. (1988) sample, the conflict practically disappears. For these stars, Jones & Walker (1988) obtain a velocity dispersion of 1.31 km s⁻¹ (adjusted to a distance of 414 pc). The velocity dispersion we find, although from a very small number of (bright) stars, is comparable to that of van Altena et al. (1988) and to the Jones & Walker (1988) bright sample. The space motions, particularly that of component C, remain a controversial subject. A comprehensive discussion can be found in O'Dell et al. (2009). Briefly, van Altena et al. (1988) find a motion of 4.8 ± 0.5 km s⁻¹ toward P.A. of 142°, significantly larger than the velocity dispersion of the ONC; Tan (2004) interpreted this velocity as evidence that the BN infrared and radio source object (which is moving in the opposite direction) originated in the OT. However, later studies of the radio sources in the embedded BN-KL cluster region (Rodríguez et al. 2005; Gómez et al. 2005) found three radio sources moving away from a common point where they coincided some 500 yr ago, and showed that the source used by Tan (2004) as reference was one of the moving objects.Photographic determination of astrometric positions of stars is notoriously difficult, and van Altena's tangential motion of 4.8 km s⁻¹ is almost certainly too high (O'Dell et al. 2009). Van Altena et al. (1988) find a poor agreement of their proper motion of component C with that found by McNamara (1976). Indeed, the motions obtained by McNamara are, in units of mas yr⁻¹, μ_x = −8.3, μ_y = −5.7, quite incompatible with those found by van Altena et al. (1988), μ_x = +1.4 ± 0.17, μ_y = −1.8 ± 0.17. The data from van Altena et al. (1988) indicate that components B and C are moving in opposite directions at about 3.5 mas yr⁻¹, a result incompatible with the relative motions of these stars found by Allen et al. (1974), who over a time interval of over 125 yr find no significant relative change in their separation. The results of the present study imply a separation velocity of component B relative to C of 0.71 mas yr⁻¹ (1.4 ± 0.9 km s⁻¹; see Figure 5 and Table 4), also incompatible with the van Altena et al. (1988) result.
• 4.  
  The rest of the components of the OT present small and random velocities that resemble the behavior of a bound and virialized system as found by Allen & Poveda (1974) and Allen et al. (1974, 2004).