For example, Bartko et al. Contamination may also affect the interpretation of the geometric structure of the disk, which was recently claimed to be highly warped Bartko et al. Further controversy exists regarding the kinematic properties of the stars that are not on the CW disk. Claims of a second, counterclockwise disk have been made Genzel et al. Precise orbital parameter estimates are necessary for resolving this issue, as the presence of a second structure has implications for both star formation and stellar dynamical evolution in the GC. The data sets and sample are presented in Section 2.
The data analysis, including image processing and astrometric and orbital analysis techniques, is detailed in Section 3. To explore the impacts of measurement error and the assumptions used in our analysis, simulations are run on mock data sets, which are presented in parallel with the observed results in Section 4. We discuss our findings in Section 5 and conclude in Section 6.
There are stars that form the sample of this study see Figure 1. These stars are selected based on the following two criteria.
Figure 1. Location of the young stars with RV and astrometric measurements that comprise the sample for this study. Sources are identified based on their astrometric properties: acceleration detections red stars , linearly moving with acceleration constraints blue squares , and linearly moving without acceleration constraints black circles.
Filled stars and filled squares mark sources with new acceleration detections and acceleration constraints, while the filled circles mark the sources with new radial velocity measurements from GCOWS. The dashed black box denotes the central 10'' field of view where the highest astrometric precision is achieved. We explicitly test this assumption in Section 5. As has been reported in previous publications Ghez et al.
From the 27 epochs of available speckle data, we use those epochs with more than frames to insure robust coordinate transformations see Section 3. Matthews in its narrow-field mode, which has a plate scale of 9. Here we include all existing Keck AO observations through , which includes 19 epochs and a time baseline of seven years Ghez et al. As compared to our previous work on the young stars in Lu et al. The observational setup was the same as the — laser guide star adaptive optics LGSAO observations reported in Ghez et al.
The images were taken at a position angle P. At least three exposures were taken at each dither position. Table 2 summarizes the narrow-field AO imaging observations used in this study. These observations were taken on May 3, May 20, and June 5. The tip-tilt star, P. In order to obtain the large FOV, we used a nine-position box pattern with a 8 5 dither offset and obtained 3—7 frames at each dither position.
For the first two epochs, we also obtained a four-position box pattern with 4'' dithers, providing large overlaps between all tiles in the mosaic. We refer to these wide-field data as "mosaics" and the details of the observations can be found in Table 3.
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For eight of these stars, this is the first report of an RV measurement in the literature. All data sets were reduced using standard data processing techniques, including sky subtraction, flat-fielding, and bad-pixel and cosmic-ray rejection. The AO data were corrected for both optical distortion using the latest solution for the NIRC2 narrow camera and achromatic differential atmospheric refraction Yelda et al.
For each observing run, individual frames are combined to make an average map. The details of this process depend on the observing technique used. The speckle data are combined to create an average image for each epoch using a weighted shift-and-add technique as described in Hornstein These images are then combined with a weighted average, where the weights are set equal to the Strehl ratio of each image.
For each epoch of mosaic data, we create an average image at each dither position i. All exposures taken at a given dither position are included in the corresponding average image except for a few cases where the frames were of extremely poor quality for one of several reasons e.
As done in our previous efforts, we create three independent subset images of equivalent quality in order to determine astrometric and photometric uncertainties for the speckle and AO central 10'' images. Likewise, subset images are created for each of the individual dither positions in the mosaics. Stars are identified and their relative positions and brightnesses are extracted from all images using the PSF fitting algorithm StarFinder Diolaiti et al.
A model PSF for each image is iteratively constructed based on a set of bright stars in the field that have been pre-selected by the user. The model PSF is then cross-correlated with the image in order to identify sources in the field. The set of PSF stars used for each image in the mosaic, on the other hand, depends on the position of that image within the wide mosaic FOV. To identify sources, we use a StarFinder correlation threshold of 0. The initial star list for each epoch contains only those sources that are detected in the average image and in all three subset images.
The inaccuracies in the PSF model for the AO images occasionally lead to spurious source detections near bright stars. There are two sources of statistical uncertainty associated with each positional measurement in the narrow-field images. Second, there is a term that appears to arise from inaccuracies in the estimates of the PSF wings of neighboring sources Fritz et al. Figure 2 shows the centroiding and additive errors for each of our speckle and central 10'' data sets.
In addition, three of the AO data sets were taken at either different positions or position angles than the rest of the AO observations and therefore are impacted by residual distortion left over after the distortion correction is applied, as described in Yelda et al. Figure 2. The median uncertainty of the young stars is reported for each epoch. Alignment errors are minimized near the reference epoch, June, and increase with time away from this epoch see Section 3. The additive errors for speckle and AO are shown as dashed lines.
The speckle observations were taken in stationary mode, and so the field rotated over the course of the night. This led to an FOV with varying numbers of frames contributing to each pixel in the final image. As a result, stars near the edges of the FOV had relatively poor astrometric measurements. Final star lists for the wide field mosaics require additional steps and a different treatment of the uncertainties.
Once the central tile is transformed, a new reference list of positions is created in the following way. For stars that are matched, their positions and their associated errors are updated. The new positions are taken as the weighted average of the positions in the existing reference list and the transformed star list. This new list then serves as the reference list for the stitching of the next tile in the sequence. This procedure is repeated until all tiles are aligned. After the central field from the nine-point dither observations is first aligned, the tiles from the four-point dither if they were taken are aligned in the order: SW, NE, SE, NW.
After completing the full alignment, we refine this intermediate star list by once again transforming each tile's star list to it a final time. In this instance, the averaging is done once all tiles are transformed and the intermediate reference list of positions is not included in the averaging. Each of the alignments performed in these steps involves a second-order polynomial transformation, which consists of 12 coefficients. In order to measure relative positions and proper motions, stellar positions from each epoch must be transformed to a common reference coordinate system.
This procedure is complicated by the fact that stars available for performing the transformation have detectable proper motions. Previous GC astrometric reference frames were constructed by minimizing the net displacement of reference stars between star lists, a procedure which implicitly assumes that these stars have no net motion over the field the "cluster" reference frame; e. Neglecting to account for this motion results in degeneracies between the transformation parameters and the measured stellar velocities.
It is therefore important to understand the motion of these stars if they are to be used in the construction of a stable reference frame. The absolute positions and proper motions presented in Yelda et al. Here we update the positions and velocities of these "secondary" standards using a slightly modified version of the analysis described in Yelda et al. Specifically, we now use mosaicked star lists as opposed to mosaicked images. The final positions and their uncertainties are computed using a similar procedure as described in Section 3.
The alignment of the stars' positions across all epochs is a multi-step process. The star lists from the deep central AO and speckle images are transformed to the coordinate system defined by the June AO image using a second-order polynomial transformation. In the alignment of each epoch, t e , we first propagate the positions of the secondary astrometric standards from t ref to the expected positions in t e using their known absolute proper motions. We then find the best-fit transformation from the measured positions in t e of the astrometric standard stars to their expected positions.
This use of velocity information allows us to use all the astrometric standards, regardless of spectral type, and removes the degeneracy between frame transformations and the stellar velocities. These alignment errors are a function of time from the reference epoch and of the number of reference stars used in the transformation. Given the high stellar density environment of the GC, it is important to consider the effects of source confusion Ghez et al.
Stellar positions can be affected by unknown, underlying sources that have not previously been detected, or they may be affected by known sources that, when passing sufficiently close to a star, get detected as only one source instead of two. While it is not possible to account for the former case, we can determine when a star's positional measurement is biased by another known source.
Using preliminary acceleration fits see Section 3. For epochs in which the predicted positions of two stars come within 60 mas of one another roughly the FWHM of our images , but only one star is actually detected, we exclude that detection as it is likely confused by the undetected source. A total of 79 positional measurements were removed due to confusion, leaving positions for the narrow-field sources combined. The mosaic star lists are aligned in a similar way as described above, but separately from the deep central and speckle data.
The reference epoch chosen for the alignment of these three star lists was the observation, as this was the mid-point of these data sets. In other words, the astrometry obtained from the narrow field data sets takes precedence over the mosaic astrometry. All the x and y positions are independently fit as a function of time with kinematic models. For the central 10'' field, each star is fit with two models: 1 proper motion only and 2 proper motion and acceleration. Stars detected beyond the central 10'' field i. The reference time, t 0 , for the position, velocity, and acceleration measurements of each star is chosen as the mean time of all epochs, weighted by the star's positional uncertainties.
The velocity fits take on the form. Whether a star has measurable accelerated motion depends on several factors, including its distance from the SMBH, the time baseline over which it is detected, and the precision with which its positions are measured. This increases the number of acceleration measurements beyond 1'' over our previous work in Lu et al. Furthermore, Gillessen et al. For all other sources, the proper motion fit is used.
We present the positions, proper motions, and accelerations for our sample in Table 4. Figure 3. Positional measurements for the six stars beyond a projected radius of 0 8 with reliable acceleration detections. Positional uncertainties do not include errors in the transformation to absolute coordinates i. The significance of each star's acceleration in the radial direction is shown in the upper right corner of each panel.
Figure 4. Residuals in X left and Y right after subtracting the best-fit acceleration curves shown in Figure 3 for each of the six accelerating sources. Note that some RV measurements reported in Bartko et al. For stars with acceleration limits, the positions and velocities are from the linear fits and the acceleration limits are from the acceleration fits.
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The position, proper motion, and acceleration uncertainties from the fitting procedure as a function of projected radius are shown in Figure 5. The observed increase in errors with radius is a result of alignment uncertainties and the number of epochs. For the central 10'' sources, the median errors in positions and proper motions are 0. The position and proper motion measurements of stars at large radii and detected in only the wide mosaics have typical uncertainties of 0. These relatively high uncertainties are a result of having only three measurements and a four-year baseline.
We also show the astrometric uncertainties as a function of K magnitude and number of epochs for stars in the central 10'' data set in Figure 6. The figure shows that the uncertainties have little dependence on magnitude but strongly correlate with the number of epochs a star is detected in. These measurements match and sometimes exceed the highest astrometric precision that has been reported to date Gillessen et al.
For completeness, we show the RV uncertainties for all young stars in the sample and indicate the source of the measurement that we use in our analysis i. Figure 5. Observed position left , proper motion middle , and acceleration right uncertainties as a function of projected radius.
The average uncertainty along the X and Y coordinates are plotted. Figure 6. Position top , proper motion middle , and radial acceleration bottom uncertainties as a function of K magnitude left and number of epochs right for our sample of young stars beyond a projected radius of 0 8 and in the central 10'' AO data set.
The astrometric uncertainties are estimated from either the proper motion or acceleration fit to each star's individual positions over time. Stars with acceleration detections and acceleration constraints are shown as open red circles and open green squares, respectively.
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The figures show that our astrometric uncertainties have only a slight dependence on stellar magnitudes and a strong dependence on the number of epochs a star was detected in. The velocities are then transformed to the local standard of rest LSR reference frame by correcting for the Earth's rotation and motion around the sun, and for the Sun's peculiar motion. Further details on the RV extraction process are reported in Ghez et al.
The RV values and their uncertainties are reported in Table 4. The 15 stars that are in common between the Do et al. The Do et al. In contrast, Bartko et al. Figure 7. Right: comparison of radial velocity measurements for the 15 common stars in the two data sets. The description of the central potential used in this analysis is based on a spherically symmetric mass, M tot , located at a distance R 0 , and composed of the mass of the central SMBH, M BH , and an extended mass component from the nuclear star cluster, M ext. If M ext is modeled as. Stars with acceleration detections and with useful upper limits are shown in Figure 8.
For stars that show significant deviations from linear motion in the plane of the sky, the measured a R is converted to a line-of-sight distance through the following relationship:. While we do not know the line-of-sight distance and therefore the full 3D distance a priori, we use the star's projected radius R as the star's 3D radius r to determine M ext , which is a lower limit on the true extended mass. This allows Equation 8 to be rearranged to obtain z ,.
We note that there is a sign ambiguity in the line-of-sight distance, which results in degenerate orbital solutions. Figure 8. The theoretical maximum acceleration a z 0 for the nominal black hole mass of 4. These sources have known line-of-sight distances and therefore have the best determined orbital solutions. With two possible z solutions in hand, two sets of orbital elements are found. The probability density functions PDFs for each are constructed by carrying out a Monte Carlo MC simulation in which 10 5 artificial data sets are created.
We note that in a given trial, all stars' orbits are determined using the same gravitational potential. The PDFs are constrained to small regions of parameter space for positive and negative z. Figure 9. Probability distribution functions for eccentricity left , inclination middle , and angle to the ascending node right as a function of the line-of-sight distance for the six stars with significant accelerations in the plane of the sky.
The absolute value of the line-of-sight distance, z , is precisely determined for each of these stars from their measured accelerations. The sign ambiguity of z results in the degenerate set of solutions. Accelerations that are consistent with zero can also provide constraints on the line-of-sight distance. Thus, the non-detection of an acceleration translates to a lower limit on the line-of-sight distance. Furthermore, the minimum acceleration allowed, a bound , is set by the assumption that the star is bound. For all other stars, including those outside the central 10'' field i.
Compared to our earlier efforts in Lu et al. Taken together, these improvements provide tighter constraints on the orbits of the young stars as well as any kinematic structures present. We construct various ensemble distributions from the real data in Section 4. We also incorporate simulations of mock data sets, which are run through our orbital analysis and combined into the same distribution functions. The results are compared to the real data in order to model the true underlying distributions and to explore any biases introduced by measurement uncertainties and assumptions in our analyses Section 4.
Kinematic structures are identified by constructing a density map of the normal vectors to the stars' orbital planes. The direction of the normal vector is described by inclination, i , and the P. These values are then averaged over all 10 5 MC trials to produce an average density map.
A clear peak of 0. Figure An overdensity of 0. While the existence of the CW disk has been well-established prior to this work, it is important to estimate disk membership probabilities for each star in order to properly characterize the disk properties. The probabilities are calculated following Lu et al. In Lu et al. The true disk fraction, however, is likely to be smaller than this and is explored below using mock data sets. Velocity vectors of all stars in the sample. The arrows are color-coded according to their disk membership probability.
The dashed circles mark the three radial bins discussed in Section 4. The stars in the CW disk are found to have non-circular orbits. In Figure 12 , we plot the solutions of the accelerating stars separately from those of the non-accelerating stars. Only orbital solutions that fall within 15 2 of the disk solution are included for each star, thus weighting the distribution by disk membership probabilities.
Below, we explore the impact and possible bias of measurement uncertainty since the eccentricity is a positive definite quantity and of the uniform acceleration prior on what is observed. Left: eccentricity distribution of the clockwise disk. All orbital solutions falling within 15 2 of the disk are included, thereby weighting the distributions by disk membership probability.
Right: eccentricity distributions shown separately for likely disk members with acceleration detections solid and without dashed. Thus, the dispersion angle of the disk does not get larger with radius. To explore the impacts of measurement error and the acceleration prior assumptions used in our orbital analysis, we construct mock data sets with known underlying kinematic properties.
Both stars with a common orbital plane i. In each set of simulations performed, we create mock kinematic data x , y , v x , v y , v z , a x , a y , add errors to each of these variables, and run our MC orbital simulations in the same way that the observed data are treated. We choose to use a point mass since including the extended mass in the analysis of the real data did not make a difference in the final results. We assume the surface density profile found by Do et al.
The distribution of orbital eccentricities for the disk stars depends on the simulation. From these simulated orbits, we select the 3D positions, velocities, and accelerations at a particular "observation" time, which we take as Mock accelerations are only determined for stars within 5'' of the black hole, consistent with our treatment of the real data. We consider only those simulated stars whose projected positions are within the FOV covered by the Keck and VLT spectroscopic observations.
The noise added to the mock data is based on the observed measurement uncertainties as a function of distance from the black hole, as astrometric uncertainties tend to increase with radius see Figure 5. We determine the minimum and maximum uncertainties in position, velocity, and acceleration of the known young stars in our sample in 1'' radial intervals. In each trial, the uncertainties assigned to a simulated star are randomly selected from a uniform distribution between these boundaries for the appropriate radial interval dependent on the simulated star's projected radius.
We then run 10 4 MC trials in which we sample from the mock data and the assigned uncertainties for each simulated star. This results in a six-dimensional PDF representing the probability distributions for the six orbital elements. For the remaining stars, a uniform acceleration prior is used, imposing the same boundaries of the minimum acceleration allowed given a bound orbit and the maximum acceleration given the star's projected radius. Table 5 summarizes the mock data sets created and we describe their details here.
Density maps were produced for all simulated data sets ID 1 in Table 5. This prior results in smaller line-of-sight distances on average than the simulated stars' true distances, and small z will favor face-on orbits over edge-on orbits. The global structure of the disk can be described by studying its orientation as a function of radius.
The radial intervals used are 0 8—3 2, 3 2—6 5, and 6 5—13 3. The significance of a density enhancement found in either of these maps is determined as in Equation 13 , but relative to 40 isotropically distributed orbits within the radial bin of interest. Density of normal vectors for stars in the three separate radial bins: 0 8—3 2 top , 3 2—6 5 middle , and 6 5—13 3 bottom. The middle radial interval shows hints of the CW disk and extended structure around this location. The same scaling is used in each plot to show the relative strength of the features. We caution, however, that this feature is a result of mainly three stars and that the outer radial bin is not sampled uniformly in azimuth.
While Bartko et al. Furthermore, the previously proposed counterclockwise disk is not detected in any radial bin in this work. We therefore conclude that the population not on the CW disk aside from the three stars with common orbital planes in the outer radial bin is consistent with an isotropic distribution within the measured uncertainties. The resulting eccentricity distributions from the orbital analysis on the simulated disks with a range of input eccentricities ID 2; see Figure 15 for two cases were compared to the observed values.
For each simulation, the generated velocity vectors of the stars are shown on the left, with the location of the black hole being marked as a red X at the center. The eccentricity distributions of the accelerating solid and non-accelerating dashed stars from each simulation are shown on the right. The orbits of the accelerating stars are more accurately determined, as expected. Based on these simulations, the observed eccentricity distribution in Figure 12 cannot be a result of measurement bias added to an intrinsically circular disk. Measurement uncertainties of the individual eccentricities can both bias the observed average values and increase the width of the eccentricity distribution.
This shows that there is a bias in the average eccentricity that is more substantial for the non-accelerating stars. Furthermore, if we treat the rms values from the simulation as a bias term and subtract in quadrature from the observed rms values, then it appears that most of the spread in the observed eccentricities can be accounted for by measurement uncertainties. We obtain a formal estimate of the intrinsic rms of 0. This is the first time the measurement bias has been quantified via simulations and explicitly accounted for in estimates of the eccentricities of stars on the CW disk.
The eccentricities are slightly lower than previously determined Beloborodov et al. Our ability to estimate the true fraction of disk members can be quantified using the orbital analysis of mock data involving a combination of disk and isotropic stars ID 3 in Table 5. The left panel of Figure 17 shows these results, averaged over the 10 trials for each disk fraction, with a second-degree polynomial fit to the data.
The method for identifying disk stars, described in Section 4.
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The right panel of Figure 17 shows the ratio of the estimated number of candidates from this method to the true number of disk members for each model, which reveals the degree of contamination from the non-members. With the conservative cut used to select candidate disk members described in Section 4. Left: sum of the squared differences in density between the disk fraction models and the observations see Section 4. Right: the level to which the true number of disk members is overestimated in each disk-fraction model, again averaged over the 10 trials run for each disk fraction model.
As a visual reference, a dashed line marks where the number of candidate disk stars equals the true number in each simulation. Note that the Y -axis is truncated for clarity. True disk members are shown as the red solid histogram, while the isotropic population is shown as the black dashed histogram.
While this conservative threshold identifies all true disk members as candidates, there is an abundance of contaminants, even at the highest probabilities. An additional isotropic stars were cut off to the left of the figure for clarity. In short, our star is in hydrostatic equilibrium—inward pressure is equal outward pressure. Next, in another five or so billion years time our star will run out of hydrogen to burn at its core.
It is at this point that helium burning begins; however, there is a step to consider before this. This is known as the Helium Flash , a thermonuclear runaway event where the entire helium core bursts into life in an instant, a massive amount of energy is released. It is, however, not enough to destroy the star. True, it will make it expand considerably as the outer parts take the brunt of the explosion, but gravity will have its way and pull the material back in again until a new equilibrium is reached. This new equilibrium will be larger than on the hydrogen burning stage, but smaller than what it achieved shortly after the helium flash.
At this point, it is important to note that variable or pulsating stars are the exception to this equilibrium. Many larger stars at some point during their lives will, for a while, become unstable. Cepheid Variables are just one type of these pulsating stars. Through all of the previous stages of a stars life, gravity keeps trying to crush the core. However, it continues to fuse and produce energy; hydrogen fuses with hydrogen to produce helium; helium fuses with helium to produce carbon and oxygen…this keeps it in hydrostatic equilibrium.
If the iron core is going to be heavier than 1. As a result, the electrons and protons fuse together to make neutrons. If the core is less than solar masses then a neutron star will be the supernova remnant, if more, a black hole will emerge. Briefly, with a 2 solar mass neutron star core, when the star runs out of all fuel the iron core shrinks dramatically to about the size of a city crushed by gravity until neutron degeneracy stops it from contracting into a black hole.
When these outer layers come crashing in on the neutron star at the center, they effectively bounce off of it and sail outwards into the universe. Is it possible for a star to explode without there being any remnant left over? Or even worse is it a triple bluff??
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