galaxies are traced by the different kinds of tracer populations we use spheroidal galaxies from the Magneticum Pathfinder Simulations Box4 uhr in Sec. 6.3. We will discuss velocity features that can be detected by observations of the tracer populations and the possible implications of those features for the formation history of a galaxy in Sec. 6.4. Finally, we conclude this chapter with a short summary and discussion in Sec. 6.5.
6.2. NUMBER STATISTICS 135
Figure 6.1: Two examples for the Line-of-Sight (LOS) velocity profiles measured by a random selection of 100 tracer stars from a simulation of a spiral galaxy.Upper left panel: Edge-on view on the simulated spiral galaxy, with stars marked as black dots. The central disk lane is masked due to the fact that observers would not be able to detect tracers in this region because of the high luminosity. Red open diamonds mark the 100 randomly chosen stars that are used as tracers. Upper right panel: LOS-velocity profiles calculated from all stars (black solid line), from all 100 tracer stars shown in the upper left panel (red solid line), and only from the tracer stars that are less than two times the scale height hsaway from the disk lane (blue dash-dotted line). The blue symbols show the LOS-velocities of the individual tracer stars colored according to their distances from the disk lane: within1hs(light blue diamonds), between1hsand2hs(medium blue stars) and farther away than2hs(dark blue circles). Lower left panel: Same as upper left panel but for a different set of randomly chosen tracers. Lower right panel: Same as upper right panel but for the tracer set shown in the lower left panel.
Figure 6.2:Left panel: Histograms for the scale heights hscalculated from the 100 repetitions of N randomly drawn tracer stars, with N=20tracers (blue line), N=50tracers (cyan line), N =100tracers (green line), N =500tracers (yellow line) and N =1000tracers (red line). Right panel: Histogram of the LOS-velocity–
RMS-test for the same sets of 100 repetitions of N randomly drawn tracers as in the left panel. The smaller the values, the better does the LOS-velocity profile estimated from the N tracers represent the true profile (see right panels of Fig. 6.1, red and black lines respectively).
to the galaxy’s plane, similar to the case shown in the lower panel of Fig. 6.1.
Tracers are often used to estimate velocity profiles for the galaxies, as long as these are not seen face-on. For a spiral galaxy, for example, a face-on view would not provide any information about the rotational velocity since the ordered motion of rotation in those cases is orthogonal to the line-of-sight (however, Herrmann et al. (2008) and Herrmann & Ciardullo (2009) studied the PN systems of 6 face-on disk galaxies to test if the mass-to-light ratio is really cface-onstant in spiral disk galaxies). For each of our tracer drawing events we calculate the resulting velocity profile and compare it to the LOS-velocity profile calculated from all stars within the studied area of our simulation. For both profiles we bin the LOS-velocities in 40 equal-distance bins of 1 kpc each along the x-axis, independently of the number of objects included in each bin. The results for the two examples discussed above are shown in the right panels of Fig. 6.1, with the “real” LOS-velocity profile shown as black solid line and the estimated profile from the 100 tracer stars shown as red solid line. The positions and LOS-velocities of the individual tracers are shown as blue open symbols, with different symbols and shades of blue according to the distance of the tracer from the galaxy’s disk plane. As can be seen in the upper panel, the tracers within the halo have significantly different velocities from the tracers that are part of the disk structures. This is due to the fact that the halo stars have non-ordered, spherical orbits, while the disk stars have ordered rotation on circular orbits around the galaxy’s center within the disk plane. Thus, excluding the tracers within the halo from the calculation of the LOS-velocity profiles give smoother and more accurate profiles in general (see the blue dash-dotted line in the upper right panel of Fig. 6.1), however, at the cost of lower number statistics.
6.2. NUMBER STATISTICS 137
We investigate the accuracy of the estimated profiles from the randomly drawn tracers compared to the “real” LOS-velocity profile using an RMS-test, by calculating the mean deviations of the tracer-profile from the real tracer-profile for each experiment. The right panel of Fig. 6.2 shows the results of these tests as histograms for the 100 experiments performed for each of the five different numbers of tracers N(N =20 (blue line),N=50 (cyan line),N=100 (green line),N =500 (yellow line) andN=1000 (red line)). The histogram clearly shows that, the more tracers are available to calculate the LOS-velocity profiles, the better the representation of the underlying general rotational LOS-velocity profile, i.e., the experiments withN =20 tracers (blue line) exhibit generally larger RMS-deviations than the experiments withN=1000 tracers (red line).
For those disk galaxies which are seen edge-on, the tracer density perpendicular to the disk can be used to estimate the scale height of the disk by fitting an exponential to the density (as has been done by Shih & M´endez (2010)). We estimated the scale height hsfor all our experiments as well, with the resulting histograms shown in the left panel of Fig. 6.2. While the 100 experiments with N = 20 tracers drawn from the full sample cover the whole range of resulting scale heights from 0.05 kpc < hs < 1.2 kpc (blue line), the 100 experiments with N = 1000 converge on values of 0.4 kpc < hs < 0.6 kpc (red line), with the real scale height hs = 0.48 kpc (black dashed line, calculated from all stars within the simulation). In general, forN=500 andN =1000 the probability of getting a good approximation of the scale height is very high, while forN = 20 andN = 50 the likelihood for a good representation of the real scale height is rather low. In the case ofN =100 tracers (green line), the resulting scale heights will most likely be a good approximation of the true value, yet there is still a high probability of too low or too high values (from 100 experiments, 68 have a scale height withinhs=0.48±0.1 kpc, and only nine have a scale height outside ofhs=0.48±0.2 kpc).
Most of the currently ongoing surveys using PNe or GCs as tracer populations, however, are studying early-type galaxies and not disk galaxies, since those systems are approximately spheroidal and thus the resulting kinematic information is less strongly dependent on the inclination angle under which the galaxy can be observed. Also, as mentioned before, the obscuration due to dust and star for-mation is much lower in early-type galaxies than in late-type galaxies, which facilitates the detections of the tracers (Blom et al., 2012a). These observations can be used together with kinematic surveys studying the innermost areas of ETGs, like Atlas3D(Cappellari et al. (2011a), see also Sec. 1.2.3 for more details on this survey) to detect kinematic twists between the inner, bright parts of ETGs and their faint outskirts that might provide informations about the formation scenarios of the ETGs (see Foster et al. (2013) and Brodie et al. (2014) for first comparisons).
Therefore, we repeat our experiments of drawing 100 times N tracers, with N = 20, N = 50, N =100,N =500 andN = 1000, from a parent sample of stars, this time using a spheroidal galaxy.
We use a spheroidal from a binary merger simulation, 11 OBH2 13 (see Sec. 2.2.1), which we evolved for about 8.5 Gyr after the first encounter, ensuring that the system could relax properly and shell and stream structures have vanished by the time we conduct our experiment. This time, we include all stars within a sphere of radiusrmax=100 kpc as parent population of tracers. We exclude stars within a sphere of 1.5R1/2 (6.3 kpc) from the parent sample to mask the innermost area where the galaxy is too bright for PNe and GCs to be detected. One example of such an experiment with N = 100 tracers is shown in the upper left panel of Fig. 6.3, with all stars of the parent population shown as black points. The 100 tracers chosen in this experiment are shown as open diamonds, with colors according to their individual LOS-velocity. Here, the blue/cyan parts of the galaxy move away from the observer, while the red/orange tracers move towards the observer. This is similar to what has been shown for example by Coccato et al. (2009) and Coccato et al. (2013).
Figure 6.3: Upper left panel: Edge-on view of a simulated spheroidal from a binary merger (11 OBH2 13, see Sec. 2.2.1), at a relaxed state at about7.5 Gyrafter the merger event, with the major axis calculated within 2 times the halfmass radius (r1/2≈4.2 kpc). The galaxy has a slight twist in its density at the outskirts, causing the tilt of the major axis. The 100 randomly drawn tracers are shown as open diamonds, with their colors according to their LOS-velocity, revealing the slight rotation of the spheroidal around its minor axis. Upper right panel: Surface density profiles calculated from all stars (black solid line) and from the 100 tracer stars (red solid line). Dashed lines show the power-law fits to the density of the total (black) and the tracer (red) profile, dash-dotted lines show the respective S´ersic fits. Lower left panel: Surface density profiles calculated from all stars (black solid line) and from one experiment of N tracers each, with N =20tracers (blue line), N=50tracers (cyan line), N =100tracers (green line), N =500tracers (yellow line) and N =1000tracers (red line). The line for N =100is the same as in the upper right panel. Lower right panel: RMS-test for the deviations between the surface density profiles estimated from the tracer populations and the true profile, for 100 repetitions each.
6.2. NUMBER STATISTICS 139
Figure 6.4: Left panel: Histograms for the slopesΓ of the power-law fits to the surface density profiles calculated from the 100 repetitions of N randomly drawn tracer stars, with N=20tracers (blue line), N=50 tracers (cyan line), N =100tracers (green line), N=500tracers (yellow line) and N=1000tracers (red line).
Right panel: Histograms for the S´ersic indices n of the S´ersic fits to the surface density profiles calculated from the 100 repetitions of N randomly drawn tracer stars, with colors as in the left panel.
From the tracer samples, we calculate surface density profiles using 10 radial bins of equal number of particles for the 100 experiments of a given number of tracersN, i.e., 0.1Nparticles per bin, with the maximum radius Rmaxthe largest radius at which a tracer has been drawn for each experiment.
The upper right panel of Fig. 6.3 shows the resulting surface density profile for one experiment with N=100 tracers (the same one for which the spatial distribution of the tracers is shown in the upper left panel of the same figure) as solid red line. The surface density calculated from all stars of the parent population is shown as solid black line. The lower left panel shows the resulting surface density profiles for one experiment of each number of tracers (N =20: blue line,N =50: cyan line,N=100:
green line (same as the red one in the upper right panel), N = 500: yellow line,N = 1000: red line) on top of the surface density calculated from all stars of the parent population (black solid line). As can be seen, for all numbers of tracers the binned surface density profiles are in good agreement in the inner parts, however, the larger radii are poorly fitted in all cases. This is due to the fact that we use equal numbers of bins, and thus loose the information in the outskirts by binning 50 or 100 tracers in one bin for N = 500 orN = 1000, respectively. The deviations from the “true” underlying surface density are much larger for the smaller samples of tracers, as is also shown in the lower right panel of the same figure where we show histograms of the RMS-deviations between the true profile and those calculated from the tracer distributions.
Since we want to get as much information from the tracers as possible, it is more natural to use larger numbers of bins when larger numbers of tracers are available, but then the RMS-errors cannot be compared between the different experiments anymore. Nevertheless, if we use equal-particle bins of 10 particles per bin, we find that the profiles are fitted generally much better the more tracers are
Figure 6.5: Left panel: 1σ-deviations from the scale height hs for the 100 experiments with each N = 20 (blue), N=50(cyan), N=100(green), N=500(yellow) and N =1000(red) versus the number of tracers N.
Middle panel:1σ-deviations from the slope of the single-power-law fitΓto the radial surface density profiles for the 100 experiments versus the number of tracers N (colors as in the left panel).Right panel:1σ-deviations from the S´ersic index n of the S´ersic fits to the radial surface density for the 100 experiments versus the number of tracers N (colors as in the left panel).
available, since the outskirts are reconstructed much more accurately. For the following analysis, we therefore use equal-particle bins of 10 tracers per bin to reconstruct the underlying information from the tracer population instead of an equal number of bins for all experiments.
Usually, surface density profiles of spheroidals are fitted by either a power-law
Σ(r)∝rΓ (6.1)
or by a S´ersic profile
Σ(r)∝e−kr1/n (6.2)
with n the so-called S´ersic index (see Sec. 1.4.1 for more details). We fit all our surface density profiles given by the different tracer populations with both theoretical profiles, as shown in the upper right panel of Fig. 6.3 for the example with N = 100 tracers as red dashed (power-law fit) and red dash-dotted (S´ersic fit) lines. The fits for the underlying parent population of stars is shown as black dashed (power-law fit) and black dash-dotted (S´ersic fit) lines.
The results for the power-law slopeΓand the S´ersic index nare shown in the left and the right panel of Fig. 6.4, respectively. The colors are coded as before for the 100 experiments withNtracers (N = 20: blue line, N = 50: cyan line,N = 100: green line, N = 500: yellow line,N = 1000: red line). Both the power-law slopeΓand the S´ersic indexnfrom the fits to the surface density profiles with large numbers of tracers (N = 500 and N = 1000) are very close to the values for the “true”
profile, for all 100 experiments, while the values forΓandnscatter strongly for the experiments with the small numbers of tracers (N =20,N=50), even if their mean values agree with the values for the true profile. The case ofN=100 is on average a good fit, but strong deviations are still possible.
Unsurprisingly, we find that the surface density profiles for the spheroidals are represented better when more tracers are available. As for the experiments with the LOS-velocity dispersion profiles for the spiral galaxy we find that a number ofN = 100 tracers already has a sufficiently high probability