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PROPER MOTIONS OF THE HH 1 JET
A. C. RagaInstituto de Ciencias Nucleares, UNAMB. ReipurthInstitute for Astronomy, Univ. of HawaiiA. Esquivel, A. Castellanos-Ram´ırez, P. F. Vel´azquez, L. Hern´andez-Mart´ınez, A. Rodr´ıguez-Gonz´alez, J. S.Rechy-Garc´ıa, D. Estrella-TrujilloInstituto de Ciencias Nucleares, UNAMJ. BallyCASA, Univ. of ColoradoD. Gonz´alez-G´omezDAFM, Univ. de las Am´ericas, PueblaA. RieraUniversitat Politecnica de Catalunya
Received September 19, 2018; accepted Year Month Day
RESUMENDescribimos un nuevo m´etodo para determinar movimientos propios de obje-tos extendidos, y un c´odigo que desarrollamos para la aplicaci´on de este m´etodo.Aplicamos este m´etodo a un an´alisis de cuatro ´epocas de im´agenes del HST de[S II] del jet de HH 1 (cubriendo un per´ıodo de ∼
20 a˜nos). Determinamos losmovimientos propios de los nudos a lo largo del jet, y hacemos una reconstrucci´onde la historia de la variabilidad de la velocidad de eyecci´on (suponiendo nudosbal´ısticos). La reconstrucci´on muestra una “aceleraci´on” de la velocidad de eye-cci´on de los nudos del jet, con velocidades mayores a tiempos m´as recientes. Estaaceleraci´on resultar´a en que los nudos que ahora observamos a lo largo del jet sejunten en ∼
450 a˜nos y a una distancia de ∼ ′′ de la fuente, cercana a la posici´onactual de HH 1. ABSTRACTWe describe a new method for determining proper motions of extended ob-jects, and a pipeline developed for the application of this method. We then applythis method to an analysis of four epochs of [S II] HST images of the HH 1 jet(covering a period of ∼
20 yr). We determine the proper motions of the knots alongthe jet, and make a reconstruction of the past ejection velocity time-variability(assuming ballistic knot motions). This reconstruction shows an “acceleration” ofthe ejection velocities of the jet knots, with higher velocities at more recent times.This acceleration will result in an eventual merging of the knots in ∼
450 yr andat a distance of ∼ ′′ from the outflow source, close to the present-day position ofHH 1. Key Words:
SHOCK WAVES — STARS: WINDS, OUTFLOWS —HERBIG-HARO OBJECTS — ISM: JETS AND OUT-FLOWS — ISM: KINEMATICS AND DYNAMICS — ISM:INDIVIDUAL OBJECTS (HH1/2) — STARS: FORMA-TION evista Mexicana de Astronom´ıa y Astrof´ısica , , ?? – ?? (2016)1. INTRODUCTIONThe HH 1/2 outflow (discovered by Herbig 1951and Haro 1952) has played a fundamental role in thestudy of collimated flows from young stellar objects(YSOs), and the associated observational and theo-retical work has been reviewed by Raga et al. (2011).This system has two bright “heads”: HH 1 (to theNW) and HH 2 (to the SE), centered on the “VLA 1”radio continuum source (Pravdo et al. 1985).The VLA 1 source also has a jet/counterjet sys-tem visible at IR wavelengths (Noriega-Crespo &Raga 2012) extending out towards HH 1 and 2. Op-tically, only the slightly blueshifted N jet (pointingto HH 1) is visible (Bohigas et al. 1985; Strom et al.1985), as shown in Figure 1. This optical feature hasbeen called the “HH 1 jet”. Apart from the papersmentioned above, a limited number of papers havestudied some of the characteristics of the HH 1 jet: • optical images and proper motions: Reipurth etal. (1993), Eisl¨offel et al. (1994), Bally et al.(2002), • radio proper motions: Rodr´ıguez et al. (2000), • infrared images: Davis et al. (2000), Reipurthet al. (2000), • infrared spectra: Eisl¨offel et al. (2000), Garc´ıaL´opez et al. (2008).Some of the most striking characteristics ofHH 1/2 are their proper motions (Herbig & Jones1981; Eisl¨offel et al. 1994; Bally et al. 2002; Harti-gan et al. 2011) and time-variability (Herbig 1969,1973; Raga et al. 1990; Eisl¨offel et al. 1994). Thefact that there are now four epochs of HST imagesof HH 1/2, covering a time span of ∼
20 yr (Raga etal. 2015a, b, c; 2016a, b, c) has allowed progress onboth of these issues.Raga et al. (2016a, b) have used the 4 epochs ofHST images to determine proper motions of HH 1and 2, finding a small acceleration for the motion ofHH 1 and a small braking for HH 2 (when comparingtheir proper motions to the ones of Herbig & Jones1981). They also used the photometrically calibratedHST images (Raga et al. 2016c) to evaluate the re-cent time-variability of the emission of HH 1 and 2(comparing their line fluxes to the ones of Brugel etal. 1981).For their study of HH 1/2 proper motions, Ragaet al. (2016a, b) explored a new method for deter-mining motions of angularly extended objects, basedon a two-step process: • convolving the frames of the different epochswith wavelets of chosen widths, • spatially fitting the peaks in the (degraded an-gular resolution) convolved frames.In the present paper, we apply this new methodto the four available epochs of HH 1/2 HST [S II]images, in order to determine the proper motionsand intensity variations of the knots along the HH 1jet (which was not studied in the papers of Raga etal. 2016a, b). We also present a detailed descriptionof the method, and describe a pipeline (written inPython) developed for applying this method to ob-servational or simulated emission map time-series.The paper is organized as follows. Section 2 re-views the methods that have been used to measureproper motions in CCD frames of HH outflows. Sec-tion 3 presents the new method for deriving propermotions and intensities of extended structures, anddescribes the Python pipeline. Section 4 describesthe proper motions of the knots along the HH 1 jet,and Section 5 the time-variability of the [S II] emis-sion. Section 6 describes the standard attempts atusing the observed proper motions to reconstruct thehistory of the time-variability of the ejection and topredict the future evolution of the ejected material.Finally, the results are summarized in Section 7.2. PROPER MOTIONS ANDTIME-VARIABILITES OF HH OBJECTSFROM CCD IMAGESAs far as we are aware, the first attempt atmeasuring positions and fluxes of condensations inCCD frames of HH objects was done by Raga et al.(1990), who analyzed H α and [O III] 5007 images ofHH 1/2. These authors found the then non-trivial re-sult that even though they had only two stars in theirCCD frames (and were therefore only able to com-pute a scaling, rotation and translation rather than a“real” astrometric calibration of the images) they ob-tained positions for the HH 1/2 condensations thatcoincided with the forward time-projection obtainedwith the photographic proper motions of Herbig &Jones (1981).Raga et al. (1990) measured the positions (andpeak intensities) of the HH 1/2 condensations by car-rying out paraboloidal fits to the emission peaks seenin the images. This kind of “peak fitting” procedure(fitting mostly either a paraboloid or a Gaussian) hasbeen extensively used for obtaining proper motionsof HH outflows (see, e.g., Eisl¨offel & Mundt 1992,1994; Eisl¨offel et al. 1994).2ROPER MOTIONS OF THE HH 1 JET 3Heathcote & Reipurth (1992) tried a differentmethod to obtain proper motions from CCD im-ages of HH outflows. In their analysis of imagesof HH 34, they defined a box (including the emis-sion of the HH 34 jet) within which they carriedout cross-correlations between pairs of images. Thismethod proved to be a major improvement in de-termining proper motions of HH outflows, as in-stead of relying on the positions of sometimes ill-defined peaks, the proper motions are determinedwith the emission within a spatially more extendedbox. This process yields a cross-correlation functionwith a much better signal-to-noise ratio (comparedto the images themselves), the peak of which canbe fitted to a many times surprising accuracy. Thiscross-correlation technique has become the standardmethod for determining proper motions of HH ob-jects (see, e.g., Curiel et al. 1997; Reipurth et al.2002; Hartigan et al. 2005; Anglada et al. 2007).The main inconvenience of the cross-correlationmethod is the fact that one has to choose boxes of ar-bitrary shapes (mostly square boxes have been used),sizes and locations so as to include features that onejudges to be well defined “entities” within the im-ages. This is of course inconvenient in images withcomplex structures of different sizes, and also some-what problematic since the determined proper mo-tions clearly depend on the “cross correlation boxes”that have been chosen.In a study of a planetary nebula, Szyskza et al.(2011) used the interesting method of covering theimages with a regular array of cross-correlation boxes(which they call “tiles”). The shifts of the peaksof the correlation functions corresponding to theseboxes then give a “proper motion map” of the wholefield (actually, a low-intensity cut-off has to be im-posed so as not to obtain random motions in boxeswith no visible emission structures). Raga et al.(2012a, 2013), applied this method to HH objects(with the implementation of the method being pre-sented in detail in the latter paper).This method of cross-correlation “tiles” has theclear advantage that one only needs to define: • a size for the tiles, • a “beginning point” at which to begin to drawone of the tiles, • a “low intensity cutoff” necessary for the propermotions to be calculated.These are of course many fewer free parameters thanthe ones involved in a “free choice” cross-correlationbox scheme. However, it is evident that there are complica-tions to this method. Two of these are that: • the tiles sometimes include only part of an ap-parently coherent structure (algorithmical ef-forts to surmount this problem are described byRaga et al. 2013), • identifiable features sometimes are shifted awayfrom a tile into neighbouring tiles in the imagepairs (so that a shift has to be applied to oneof the images before applying the division intotiles, see Raga et al. 2012a).3. MEASURING PROPER MOTIONS OF HHOBJECTS WITH A “WAVELETTECHNIQUE”: A PIPELINEIn order to try to avoid these problems, Raga etal. (2016a, b) proposed (and used) an alternative,two-step method: • convolving the images with a wavelet of a chosensize, • determining proper motions from spatial fits tothe peaks in the convolved maps.This method is of course a “peak fitting method”,but it also incorporates a spatial averaging (ob-tained through the convolution with a wavelet func-tion) such as is obtained with the “cross correlationmethod”. The only free parameter of this methodbasically is the half-width σ of the wavelet function(and of course, the choice of which peaks are identi-fied as “pairs” in two different epochs!).Convolving an image with a wavelet of half-width σ has 3 effects:1. improving the signal-to-noise ratio at the ex-pense of spatial resolution,2. eliminating emitting structures with scales < σ ,3. eliminating structures with scales > σ .If one convolves images with functions similar toinstrumental “point spread functions” (e.g., with aGaussian), one eliminates small scale structures, butlarger scale structures in the images still remain. Itis, however, unlikely that proper motions determinedon images convolved with Gaussians would be sub-stantially different from proper motions measured onconvolutions with wavelets. We prefer convolutionswith wavelets basically because of the mathematicalproperties of wavelet decompositions, which allowpartial rebuildings of images with arbitrary ranges RAGA ET AL. Fig. 1. [S II] image taken with the HST in 2007 of the region including the HH 1 (VLA 1) source, the HH 1 jet andHH 1 itself. The Cohen-Schwartz (CS) star is also labeled. This image (displayed with a logarithmic colour scale) hasbeen rotated clockwise by 37 ◦ so that the axis of the outflow is parallel to the abscissa. The white box encloses theregion around the HH 1 jet shown in Figures 2-4. of spatial scales (see, e.g., Kajdic et al. 2012). How-ever, this feature is not used in the present propermotion determinations.The choice of the particular form of the waveletfunction does not affect the obtained results in a sub-stantial way. In our implementation, we have chosena “Mexican hat” wavelet: g σ ( x, y ) = 1 πσ (cid:18) − x + y σ (cid:19) e − ( x + y ) /σ , (1)where σ is the half-width of the central peak. Thisfunction has an approximately Gaussian centralpeak, surrounded by a negative ring (such that itsspatial integral is zero). Together with the “Frenchhat” wavelet, this is one of the standard “wavelet ker-nels”. For an astronomically oriented discussion ofthe properties of these wavelet kernels (together withgraphic depictions) see, e.g., Rauzy et al. (1993).The convolved maps I σ are then calculatedthrough the usual integral I σ ( x, y ) = Z Z I ( x ′ , y ′ ) g σ ( x − x ′ , y − y ′ ) dx ′ dy ′ , (2)where I ( x ′ , y ′ ) is the original (i.e., not convolved) im-age, and ( x, y ) are the coordinates of the convolvedimage. The convolutions are carried out with a stan-dard, “Fast Fourier Transform” method.On the convolved image, we then carry outparaboloidal fits to intensity peaks, from which wedetermine the positions and intensities of the peaks.From the shifts of the positions between successiveepochs, we then determine proper motions.We have developed a pipeline (written in Python)that: 1. reads an image,2. convolves it with Mexican hat wavelets of thespecified σ values,3. finds peaks (either chosen by the user, orsearches for all peaks above a given intensitythreshold) and carries out paraboloidal fits,4. has a “user confirmation and labelling” routine(with which the user can choose the relevantpeaks),5. identifies the same peaks in two or more imagesand calculates the proper motions (with linearleast squares fits to the knot positions as a func-tion of time).Item number 5 allows for several possibilities: • the more straightforward one is to calculateproper motions for the knots identified by theuser with the same label in the available epochs.This is of course appropriate for images with asmall number of emitting knots, • to automatically associate the “nearest knots”detected in two successive epochs, • to search for the nearest knot (in the followingepoch), but only in the general direction awayfrom the outflow source.It is also possible to use the wavelet spectrum ofthe individual knots in order to find the knot pairsthat are morphologically closer to each other, and tothen use the identified pairs for calculating properROPER MOTIONS OF THE HH 1 JET 5 Fig. 2. The HH 1 jet in the four available epochs of[S II] HST images (see Section 3). The labels used forthe knots (some of them not visible in all of the epochs)are given. The bottom bar gives the logarithmic colourscale (in erg cm − s − arcsec − ). A flat background (of3 × − erg cm − s − arcsec − for the first three epochsand of 10 − erg cm − s − arcsec − for the 2014 frame)has been subtracted. The boxes have a 30 ′′ horizontalextent. motions. This kind of “morphological evaluation”using wavelet spectra has been studied in detail byMasciadri & Raga (2004, in the context of search forexoplanets), but has not yet been implemented inour pipeline.Finally, our Python pipeline has routines to pro-duce appropriately labeled plots for publication.Figures 1-4 (see the following Sections) were pro-duced with these routines. After further testing andimprovements to the user interface, the routine willbe available to the community. Fig. 3. The four epochs of [S II] HST images (see Figure2) convolved with a Mexican hat wavelet of half-width σ = 4 pix (see the text).
4. PROPER MOTIONS OF THE HH 1 JETWe have taken the 4 epochs of [S II] HST imagesof HH 1/2 described by Raga et al. (2016a, b, c)obtained in 1994.61, 1997.58, 2007.63 and 2014.63(we have not analyzed the H α frames because theHH 1 jet is very faint in this line). Figure 1 shows aregion of the 1997 frame including the position of theVLA 1 source, HH 1 and the HH 1 jet. The Cohen-Schwartz (CS) star, despite its strategic location, isapparently not associated with the outflow.The analysis presented in this paper is restrictedto the region around the HH 1 jet shown with a whitebox in Figure 1. The [S II] emission within this re-gion in the four epochs is shown in Figure 2, with theknots labeled with identifications that correspond tothe ones of Reipurth et al. (2000) and Hartigan etal. (2011). Also, we have labeled knot B of theHH 501 jet (see Bally et al. 2002) with a lower case“b”. This outflow appears to have been ejected by RAGA ET AL.another source in the vicinity of the HH 1/2 source(Bally et al. 2002).In Figure 3, we show the four [S II] frames afterconvolution with a σ = 4 pix wavelet (i.e., with acentral peak with a full width of 0 ′′ . ′′ . v x ) and across ( v y ) the outflow axis with reason-able errors (ranging from ∼ →
40 km s − , seeTable 1).5. THE INTENSITIES OF THE KNOTS IN THEHH 1 JETFigure 5 shows the peak [S II] intensities for knotsA-I of the HH 1 jet in all epochs, obtained throughparaboloidal fits to the peaks of the convolved im-ages (see Section 4). It is clear that for distancesfrom the VLA 1 source larger than x ∼ ′′ there is ageneral trend of decreasing intensities as a functionof x . This trend approximately follows a I ∝ x − power law (shown with a dashed line in Figure 5).Our observations do not show in a conclusiveway that individual knots have intensities that “slidedown” the x − slope as a function of time. This is be-cause in the 2014 frame (intensities shown with open circles in Figure 5) we obtain systematically largerintensites for all knots than in the 2007 frame. Thisis a result of the fact that the HH 1 jet region hasa relatively strong reflection nebula, with peak in-tensities aligned with the jet. This reflection nebulahas a stronger contribution in the 2014 frame, whichwas obtained with the WFC3 camera (with a [S II]filter of 118 ˚A width). The first three epochs wereobtained with the WFPC2 camera (with a [S II] fil-ter of 47 ˚A width), and have a smaller contributionfrom the reflection continuum.The observed I [ S II ] ∝ x − dependence for largedistances along the HH 1 jet (see Figure 5) is in re-markable agreement with the prediction of the ana-lytic, “asymptotic regime” of periodic internal work-ing surfaces of Raga & Kofman (1992). These au-thors note that at large enough distances from thesource, the decaying working surfaces should havean intensity I ∝ x − ( κ +1) , where κ is the index ofan assumed power law dependence of the line emis-sion as a function of shock velocity (see equation25 of Raga & Kofman 1992). If one takes the plane-parallel shock models of Hartigan et al. (1987), fromthe lower range of the shock velocities of their mod-els one obtains that the [S II] intensity has a scal-ing ∝ v − shock . Therefore, the asymptotic regime ofRaga & Kofman (1992) then predicts a [S II] inten-sity ∝ x − , in surprisingly good agreement with ourobservations of the HH 1 jet (see Figure 5).6. THE PAST AND FUTURE EVOLUTION OFTHE HH 1 JETAs can be seen in Table 1, along the HH 1 jetwe see a general trend of decreasing velocities withincreasing distances from the outflow source. Sucha decreasing velocity trend could in principle be theresult of drag due to entrainment of stationary, en-vironmental material.It is clear that in some of the “parsec scale HHjets” (e.g. in HH 34, see Devine et al. 1997) a pro-gressive decrease in proper motions for the “heads”at larger distances are seen, and that this trend can-not be explained as a result of a secularly increasingejection velocity from the outflow source (Cabrit &Raga 2000). This rather dramatic slowing down ofthe HH 34 “heads” is due to the fact that a precessionof the outflow axis results in a direct interaction ofthe successive heads with undisturbed environmentalmaterial (Masciadri et al. 2002).As the knots along the HH 1 jet are very wellaligned, we would not expect them to slow down dueto frontal interaction with the surrounding, station-ary environment (as occurs in the giant HH 34 jet,ROPER MOTIONS OF THE HH 1 JET 7TABLE 1PROPER MOTIONS OF THE HH 1 JET knot v xa v ya [km s − ]A 128 (91) 86 (45)B 245 (12) − −
24 (5) a the values in parenthesisare the estimated errorsFig. 4. 2007 [S II] image with the proper motions derived from the four available epochs (see Table 1). The scale of thevelocity arrows is given by the arrow in the top left corner of the plot. see above). One might still have “side entrainment”into the HH 1 jet, resulting in some amount of slow-ing down at increasing distances from the source.This effect has recently been evaluated by Raga(2016, in terms of a somewhat uncertain “ α prescrip-tion” for the entrainment velocity), who finds thatin order to obtain a substantial slowing down oneneeds a surrounding environment (in contact withthe jet beam) ∼
10 to 100 times denser than the jet.This is unlikely to be the case in the optically visibleHH 1 jet, which has already emerged from the densecore surrounding the outflow source. Also (as dis-cussed by Raga 2016), the effect of buoyancy (whichincludes the gravity and the environmental pressuregradient) is negligible for the high velocities of HHoutflows.We therefore interpret the decreasing proper mo- tion velocities (with increasing distances from theoutflow source) along the HH 1 jet as ballistic mo-tions resulting from an increasing ejection velocityas a function of time. We then take the positions ofthe HH 1 jet knots in the 2007 frame, and calculatethe dynamical ejection times t dyn = − xv , (3)where x is the distance from the outflow source and v is the proper motion velocity. In Figure 6, we thenplot v as a function of t dyn , which is the “ballisticknot” prediction of the past ejection time-variabilityhistory of the outflow source. The ejection velocityhas a general trend of increasing velocities towardsmore recent times, which we fit with a straight line, RAGA ET AL. Fig. 5. Peak [S II] emission of the knots (determinedfrom the fits to the convolved maps of the four epochs)as a function of distance from the VLA 1 outflow source.The points corresponding to the four epochs are shownwith different styles of dots (as specified in the text abovethe plot) and the successive knots are joined by lines ofdifferent colours (with labels in the same colour givingthe identifications of the knots). giving: u ( τ ) = (303 ±
15) + (0 . ± . τ , (4)where u ( τ ) is the ejection velocity in km s − and τ is the ejection time in years ( τ = 0 correspondingto 2007, since we have used the knot positions ofthis epoch). The linear least squares fit (equation4) has been calculated with the method described inAppendix A.The ejection velocity clearly should also have ashort-term variability that produces the knots thatwe observe along the HH 1 jet, so that the trendof equation (4) would actually correspond to a long-term variability superimposed on the “knot produc-ing” mode (see, e.g., Raga et al. 2015c).An interesting question is whether the relativelylow velocity of knot H (see Table 1 and Figure 6)is evidence that the more recently ejected, opticallydetected material is starting to show a decreasingejection velocity vs. time trend. Given the largeerror of our H knot proper motion (see Figure 6), itis hard to conclude that this is indeed the case.Also, we can obtain a second estimation of themotion of knot H as follows. We take the separationbetween knots H and F in the 1998.15, [Fe II] 1.64 µ mimage of Reipurth et al. (2000) (in which not H isalready visible), and compare it with the separationbetween these two knots in our 2014.63. From thiscomparison, we find that knot H has an axial motion Fig. 6. Proper motion velocities of the knots as a functionof dynamical ejection times (calculated through equation3, with t dyn = 0 corresponding to 2007). The straightline corresponds to the linear fit given in equation (4). ± − faster than the motion of knot F. Com-bining this result with our knot F proper motion (seeTable 1), we obtain a (276 ±
18) km s − velocity forknot H, which is consistent (within the errors) withthe proper motion obtained from the optical images(see Table 1), but does not support the existence ofa drop in ejection velocity associated with this knot.In order to use the present day positions andproper motions to predict the future evolution of theHH 1 jet, in Figure 7 we plot the ballistic trajectoriesof the knots on a ( x, t ) plot (where x is the positionof the knots as a function of time t ). The t = 0 axiscorresponds to the 2007 knot positions. For knot H(the trajectory with the smallest x at t = 0), wehave used the 276 km s − velocity estimated fromIR images (see above).In Figure 7, we see that the knot trajectories havecrossing points in the x = 0 → ′′ distance range, attimes smaller than ∼
500 yr. Therefore, by the timethat the HH 1 jet knots have reached the present-day position of HH 1 (also shown in Figure 7), manyknot-merging events will have occurred. This resultis similar to the one found by Raga et al. (2012a)for the HH 34 jet.In order to visualize the effect of the knot-merging events, we use the simple momentum con-serving knot-merging model of Raga et al. (2012b).We take the 2007 HH 1 jet knot positions and veloc-ities, and assign equal masses to all knots. We thenfollow the knot trajectories, merging colliding knotsusing mass and momentum conservation conditions.The knots are not assigned sizes, so that knot colli-sions take place at the points of trajectory crossingsROPER MOTIONS OF THE HH 1 JET 9
Fig. 7. Ballistic trajectories of the HH 1 jet knots inan ( x, t )-plane (where x is the distance from the out-flow source and t is the time measured from 2007). Thepresent-day position of HH 1 is indicated on the bottomright of the plot. (see Figure 7).The result of this exercise is shown in Figure 8.In this figure we show the knot positions at 150 yrintervals ( t = 0 corresponding to 2007). The knotsare represented as circles (centred on the knot po-sitions) with radii proportional to the mass of theknots. It is clear that by t ∼
450 yr most of theknots have merged, and that at this time the posi-tion of the merged knots is slightly upstream of thepresent position of HH 1 (at x ≈ . × cm or75 ′′ at a distance of 400 pc). Therefore, the materialthat is being ejected now (in the HH 1 jet) will even-tually form a new “head” close to the present-dayposition of HH 1. 7. SUMMARYWe present a discussion of the methods that havebeen used in the past for measuring proper motionsof HH outflows (Section 2), and a description of anew method that has been recently developed (Sec-tion 3). This method has been implemented in aPython pipeline.We then use this pipeline to determine propermotions of the knots along the HH 1 jet in the fouravailable epochs of [S II] HST images (obtained in1994, 1997, 2007 and 2014). We find proper motionsthat are well aligned with the outflow axis, and withvalues ranging from ∼ →
290 km s − , with thefaster velocities mostly in the knots closer to the out-flow source (see Section 4 and Table 1).For each knot, we calculate a dynamical ejectiontime, and then plot the outflow (proper motion) ve- Fig. 8. Results of the “momentum conserving knot”model described in section 6. The t = 0 (top) frameshows the 2007 positions of the HH 1 jet knots, and thefollowing frames show the knot positions at 150 yr timeintervals. The radii of the circles (indicating the knotpositions) are proportional to the mass of the mergedknots. The present-day position of HH 1 is shown withthe dashed, vertical line. locity as a function of ejection time (see Figure 6).This plot shows that (under the assumption of ballis-tic knot motions) the outflow velocity has increasedtowards more recent times.This “acceleration” of the ejection (as a functionof ejection time) implies that the knots presently ob-served along the HH 1 jet will merge into a largeworking surface. This is seen in the crossings of theballistic knot trajectories (see Figure 7) and in themomentum/mass conserving, “merging knot” modelshown in Figure 8. From this model, we see thatin ∼
450 yr most of the HH 1 jet knots will havemerged, and that at this time the position of themerged knots is slightly upstream of the present po-sition of HH 1 (at x ≈ . × cm or 75 ′′ at adistance of 400 pc). This result is qualitatively con-sistent with the suggestion of Gyulbudaghian (1984)that the diverging proper motions of the condensa-tions of HH 1 directly imply that it was formed notfar upstream from its present-day position.This kind of morphology (a large working surfaceat large distances, and a short chain of knots thatwill merge at the position of the present-day largeworking surface) is to be expected from models of0 RAGA ET AL.two-mode ejection variabilities (see the analytic dis-cussion of Raga et al. 2015c). Also, a qualitativelymost similar situation has been previously found forthe HH 34 outflow (see Raga et al. 2012a, b), inwhich the knots along the jet will merge when theyreach the present-day position of HH 34S.An important question is whether or not this kindof configuration (of knots along a jet predicted tomerge at the present-day position of a large “head”)is found in other HH jets. For some HH outflows,it is possible that proper motion data of sufficientaccuracy might be already available, and a detailedstudy of the available data might yield interestingresults. In other cases, future observations might benecessary in order to resolve this question.Support for this work was provided by NASAthrough grant HST-GO-13484 from the Space Tele-scope Science Institute. ARa acknowledges supportfrom the CONACyT grants 167611 and 167625 andthe DGAPA-UNAM grants IA103315, IA103115,IG100516 and IN109715. ARi acknowledges supportfrom the AYA2014+57369-C3-2-P grant. We thankthe anonymous referee for constructive comments.REFERENCES Anglada, G., L´opez, R., Estalella, R., Masegosa, J., Ri-era, A., Raga, A. C. 2007, AJ, 133, 2799Bally, J., Heathcote, S., Reipurth, B., Morse, J., Harti-gan, P., Schwartz, R. 2002, AJ, 123, 2627Bohigas, J., Torrelles, J. M, Echevarr´ıa, J. et al. 1985,RMxAA, 11, 149Brugel, E. W., B¨ohm, K. H., Mannery, E. 1981, ApJS,47, 117Cabrit, S., Raga, A. C. 2000, A&A, 354, 959Curiel, S., Raga, A. C., Raymond, J. C., Noriega-Crespo,A., Cant´o, J. 1997, AJ, 114, 2736Davis, C. J., Smith, M. D., Eisl¨offel, J. 2000, MNRAS,318, 747Devine, D., Bally, J., Reipurth, B., Heathcote, S. 1997,AJ, 114, 2095Eisl¨offel, J., Mundt, R. 1992, A&A, 263, 292Eisl¨offel, J., Mundt, R. 1994, A&A, 284, 530Eisl¨offel, J., Mundt, R., B¨ohm, K. H. 1994, AJ, 108, 1042Eisl¨offel, J., Smith, M. D., Davis, C. J. 2000, A&A, 359,1147Garc´ıa L´opez, R., Nisini, B., Giannini, T., Eisl¨offel, J.,Bacciotti, F., Podio, L. 2008, A&A, 487, 1019Gyulbudaghian, A. L. 1984, Ap, 20, 75Haro, G. 1952, ApJ, 115, 572Hartigan, P., Raymond, J. C., Hartmann, L. W. 1987,ApJ, 316, 323Hartigan, P. et al. 2011, ApJ, 736, 29Hartigan, P., Heathcote, S., Morse, J., Reipurth, B.,Bally, J. 2005, AJ, 130, 2197Heathcote, S., Reipurth, B. 1992, AJ, 104, 2193 Herbig, G. H. 1951, ApJ, 113, 697Herbig, G. H. 1969, Comm. of the Konkoly Obs., No. 65(Vol VI, 1), p. 75Herbig, G. H. 1973, Information Bulletin on VariableStars, 832Herbig, G. H., Jones, B. F. 1981, AJ, 86, 1232Kajdic, P., Reipurth, B., Raga, A. C., Bally, J., Walawen-der, J. 2012, AJ, 143, 106Masciadri, E., de Gouveia Dal Pino, E., Raga, A. C.,Noriega-Crespo, A. 2002, ApJ, 580, 950Masciadri, E., Raga, A. C. 2014, 611, L137Noriega-Crespo, A., Raga, A. C. 2012, ApJ, 750, 101Pravdo, S. H., Rodr´ıguez, L. F., Curiel, S., Cant´o, J.,Torrelles, J. M., Becker, R. H., Sellgren, K. 1985, ApJ,293, L35Raga, A. C. 2016, RMxAA, 52, 311Raga, A. C., Barnes, P. J., Mateo, M. 1990, AJ, 99, 1912Raga, A. C., Kofman, L. 1992, ApJ, 390, 359Raga, A. C., Reipurth, B., Cant´o, J., Sierra-Flores, M.M., Guzm´an, M. V. 2011, RMxAA, 47, 425Raga, A. C., Noriega-Crespo, A., Rodr´ıguez-Gonz´alez,A., Lora, V., Stapelfeldt, K. R., Carey, S. J. 2012a,ApJ, 748, 103Raga, A. C., Rodr´ıguez-Gonz´alez, A., Noriega-Crespo,A., Esquivel, A. 2012b, ApJ, 744. L12Raga, A. C., Noriega-Crespo, A., Carey, S. J., Arce, H.G. 2013, AJ, 145, 28Raga, A. C., Reipurth, B., Castellanos-Ram´ırez, A., Chi-ang, Hsin-Fang, Bally, J. 2015a, ApJ, 798, L1Raga, A. C., Reipurth, B., Castellanos-Ram´ırez, A., Chi-ang, Hsin-Fang, Bally, J. 2015b, AJ, 150, 105Raga, A. C., Rodr´ıguez-Ram´ırez, J. C., Cant´o, J.,Vel´azquez, P. F. 2015c, MNRAS, 454, 412Raga, A. C., Reipurth, B., Esquivel, A., Bally, J. 2016a,AJ, 151, 113Raga, A. C., Reipurth, B., Vel´azquez, P. F., Esquivel, A.,Bally, J. 2016b, AJ, 152, 186Raga, A. C., Reipurth, B., Castellanos-Ram´ırez, A.,Bally, J. 2016c, RMxAA, 52, 347Rauzy, S., Lachi`eze-Rey, M., Henriksen, R. N. 1993,A&A, 273, 357Reipurth, B., Heathcote, S., Roth, M., Noriega-Crespo,A., Raga, A. C. 1993, ApJ, 408, L49Reipurth, B., Heathcote, S., Yu, K. C., Bally, J.,Rodr´ıguez, L. F. 2000, AJ, 120, 1449Reipurth, B., Heathcote, S., Morse, J., Hartigan, P.,Bally, J. 2002, AJ, 123, 362Rodr´ıguez, L. F., Delgado-Arellano, V. G., G´omez, Y. etal. 2000, AJ, 119, 882Szyszka, C., Zijlstra, A. A., Walsh, J. R. 2011, MNRAS,416, 715Strom, S. E., Strom, K. M., Grasdalen, G. L., Sellgren,K., Wolff, S., Morgan, J., Stocke, J., Mundt, R. 1985,AJ, 90, 2281York, D. 1966, Can. J. Phys., 44, 1079
ROPER MOTIONS OF THE HH 1 JET 11A. C. Raga & A. Castellanos-Ram´ırez, P. F. Vel´azquez, L. Hern´andez-Ram´ırez, A. Rodr´ıguez-Gonz´alez, J.S. Rechy-Garc´ıa, D. Estrella-Trujillo: Instituto de Ciencias Nucleares, Universidad Nacional Aut´onoma deM´exico, Ap. 70-543, 04510 D. F., M´exico ([email protected])B. Reipurth: Institute for Astronomy, University of Hawaii at Manoa, Hilo, HI 96720, USAJ. Bally: Center for Astrophysics and Space Astronomy, University of Colorado, UCB 389, Boulder, CO 80309,USAD. Gonz´alez-G´omez, DAFM, UDLAP, Ex Hda. Sta. Catarina M´artir, C:P. 72810, Puebla, M´exicoA. Riera: Departament de F´ısica i Enginyeria Nuclear, EUETIB, Universitat Polit´ecnica de Catalunya, Comted’Urgell 187, E-08036 Barcelona, Espa˜na ′ V X VS. T DY N
DEPENDENCEIn Figure 6, we see that the errors in v x are prob-ably more important than the errors in t dyn , so thata traditional linear, least squares fit (in which theerrors in the measured values of the abscissa are as-sumed to be zero) is probably reasonable. However,in order to obtain a more convincing result, we havedone a fit in which the errors in both v x and t dyn areconsidered.One could in principle use a standard “errors inthe two variables” least squares fit approach (see,e.g., the classical paper of York 1966), but these solu-tions are based on the assumption that the errors inthe two variables are statistically independent fromeach other, and this is not the case in our “ v x vs. t dyn ” problem.Given the fact that the error in the position x ofthe knots is much smaller than the errors in the cor-responding values of v x , the errors in t dyn are givenby: ǫ ( t dyn ) = t dyn v x ǫ ( v x ) , (5)which can be obtain by putting a perturbed veloc-ity and appropriately linearizing equation (3), or al-ternatively by using the standard error propagationrelation.We then proceed in the standard way, writingthe “true” values of the measured points as [ t dyn + ǫ ( t dyn ) , v x + ǫ ( v x )], through which the linear fit goesthrough, so that v x + ǫ ( v x ) = a [ t dyn + ǫ ( t dyn )] + b , (6)where a and b are the parameters of the linear fit thatwe want to calculate. Combining equations (5-6) wethen find: ǫ ( v x ) = at dyn − b − v x − atv x . (7)These are the deviations from the straight line fitresulting from the displacements due to the errors inboth v x and t dyn .We then define a weighted χ as: χ = X i [ ǫ ( v x ) i ] w i , (8)where the ǫ ( v x ) i values are calculated from equa-tion (7) with all of the observationally determined( t dyn , v x ) pairs, and w i = 1 /σ i , where σ i are theestimated errors of the v x values (see Table 1).Now, given the measured ( t dyn , v x ) pairs, it isstraightforward to find the values of a and b that give the minimum χ (we do this by exploring nu-merically a range of values for a and b ). It is alsopossible to estimate the errors of a and b by per-turbing (with their estimated errors) the variablesof each of the measured points, recalculating the re-sulting a and b values, and then using the standarderror propagation formula.When this method is applied to the knots of theHH 1 jet, one obtains a = (0 . ± .
19) yr and b = (303 ±
15) km s − (these are the values given inequation 4). These values are actually very similarto the a = (0 . ± .
18) yr and b = (299 ±
14) km s −1