3. PROPUESTA DE UNA ESTRATEGIA ADMINISTRATIVA PARA LA
3.2. Propuesta para la implantación del modelo de competencias en SNM
A model of the LMC disk is proposed by van der Marel et al. (2002), who took a sample of carbon stars located in the periphery of the galaxy and used these data to fit a model of line of sight velocity. The line of sight velocity of a star is a function of the angular distance on the sky from the centre of the disk structure ρ, and the position angle from North Φ. Disk plane orbits are modelled by a rotation curve V(R0) which is parameterised as,
V(R0) =V0 Rη Rη+Rη 0 4.1 This is the same LMC model as employed by Alves and Nelson (2000), with geometric corrections for the large spherical angle subtended by the galaxy on the sky, and includes a term for precession of the disk inclination.
The transverse centre of mass velocity vt can be expressed in components along the line of nodes, vtc = vtcos(Θt−Θ) and perpendicular to the line of nodesvts=vtsin(Θt−Θ), where Θtis the angle of the transverse velocity, and Θ is the angle of line of nodes, from North. Along the line of nodes the position angle of a star Φ is the same as the line of nodes so Φ−Θ = 0. We can define a systematic motion corrected velocity,vlon≡vlos−vsyscosρ, which yeilds the simplified relation, given sin Φ−Θ = 0 and cos Φ−Θ =±1,
vlon=vtcsinρ−V(R0)fsini =vtcsinρ−V(R0) sinρsini = sinρ(vtc−V(R0) sini)
showing that along the line of nodes the LOS velocity (corrected for systemic LOS motion) is simply proportional to sinρ. That is, if the LMC rotation curve V(R0) is linear, which it has been found to be up to 4 kpc from the centre, after which it flattens out to at least 60 km s−1(Alves and Nelson 2000).
At the centre where ρ= 0 the line of sight velocity of the disk rotation is zero, so measurements of radial velocities here measure the systemic line of sight velocity of the galaxy directly. The only systematic source of error would be the choice of rotation centre.
The disk plane velocities were calculated from our radial velocities using the model of van der Marel et al. (2002) as follows:
Vdisk(ρ,Φ) =[vsyscosρ−vtsinρcos(Φ−Θt) +D0(di/dt) sinρsin (Φ−Θ)
−vlos]×g−1 Where g is,
a function of the geometric factor,
f ≡ cosisinρ−sinisinρsin(Φ−Θ)
[cos2icos2(Φ−Θ) +sin2(Φ−Θ)]1/2
which describes the projection of the circular disk orbital velocity into the plane of the sky. Perpendicular to the line of nodes, cos (Φ−Θ) goes to zero at the rotation centre, makingg(f) small. For g≤0.2 the projection into the line of sight of disk velocity is of the order of the error in the radial velocity. We therefore exclude these data following Olsen and Massey (2007). The subset of the data employed with|g|larger than 0.2 is shown in Figure 4.16.
Our data at the centre of the LMC provide weak constraints on the global orientation of the disk. We take disk geometry parameters from van der Marel et al. (2002). The centre of mass is given asαCM = 5h27m.6 andδCM = 69.◦87. We take the line of nodes of the disk as 130◦, and the inclination angle of the disk to be 34.◦7±6◦.2 (van der Marel et al. 2002). A recent study of Cepheid and RR Lyrae standard candles to create a 3D map of the LMC arrives at an inclination of 32◦±4◦ (Haschke et al. 2012). They also find line of nodes to be 115◦±15◦.
The values for proper motion are taken from Piatek et al. (2008) with trans- verse velocity of 476 km s−1 in a direction 78◦. They find the precession and nutation terms to be consistent with zero, and we employ this result. How- ever we note that the average of van der Marel et al. (2002) and Olsen et al. (2011) gives di/dt= -0.5162 mas /yr. For D0 = 50.1 kpc this translates to
−122.6 km s−1. Thedi/dtprecession term has no effect at the very centre of the
galaxy, and up to a maximum of about 6 km s−1at the extrema of our observed
fields.
We take an iterative approach to fitting a rotation curve to the observed ve- locities. For all samples we exclude velocities with errors greater than 20 km s−1.
We estimate a systemic velocity by first assuming a model for the rotation curve. This allows us to transform the data to the plane of the LMC disk. We then use this transformed data to get a better model, then use this model to get a better systemic velocity. We show that this bootstrap method is insensitive to starting conditions.
We use the heliocentric radial velocity to estimate the systemic velocity using a very simple solid body rotation curve. This is not unreasonable in the inner 2.5 kpc of galaxy, at greater radii we expect the rotation curve to flatten out. From exploratory analysis we set the linear relation to 24 km s−1kpc−1 and assume the disk velocity is zero at the centre. For each star in our sample the systemic velocity is calculated and the distribution is analysed. The distribution is close to the Normal distribution, and the mean value of systemic velocity is 250 km s−1 and median 251 km s−1.
Using this estimate of the systemic velocity we proceed to transform the heliocentric radial velocities to in disk plane velocities. The disk velocity data are grouped by radius to give equal number bins. The van der Marel et al. (2002) model was fitted to the mean values of the binned data from our sample using a
4.3. RESULTS 1.45 1.40 1.35 1.30 −1.24 −1.23 −1.22 −1.21 −1.20 −1.19 R.A. DEC.
Figure 4.16: The velocity in the disk plane becomes sensitive to error perpendic- ular to the line of nodes at the rotation centreαCM = 5h27m.6 andδCM= 69.◦87 (van der Marel et al. 2002). Velocities in this region have almost no component in the line of sight. The subset of data which can be transformed to disk plane velocities is shown. The solid points are a subset with heliocentric radial ve- locity of 248 km s−1, a direct measurement of the systemic velocity of the LMC
0.0 0.5 1.0 1.5 2.0 2.5 3.0 −20 0 20 40 60 Disk radius kpc Disk v elocity km s − 1 0.0 0.5 1.0 1.5 2.0 2.5 3.0 −20 0 20 40 60 −20 0 20 40 60
(a) Bins have even numbers of stars, the solid line is our model. The dotted line linear fit to the outer five data bins, ex-
cluding<1 kpc low velocity points, would
result in a higher systemic velocity.
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 20 40 60 80 Disk radius kpc Disk v elocity km s − 1
(b) The Z03 data transformed to disk plane velocities and binned the same as our data. The rotation curve is very dif- ferent at the centre.
Figure 4.17
non-linear least squares method. We found parametersV0= 57 km s−1,ν= 2.5
and characteristic disk radiusR0= 1.0 kpc. The fit is plotted in Figure 4.17. The
error bars represent the error in the estimation of the mean value in the equal number bins, they do not represent the variability in the individual velocities.
Rather than the first simple linear model, we now use this more refined model to again estimate the systemic velocity from the measured heliocentric velocities. Such a distribution is shown in Figure 4.18a.
There is some degeneracy between the systemic velocity and the rotation curve model parameters, as we must assume model parameters to get an esti- mate of systemic velocity which we then plug back into our model estimation. Exploration of a range of disk model parameters show the starting systemic ve- locity arrived at does not depend sensitively on choice of model. For 2< η <3, for 40 km s−1 < V
0 < 100 km s−1, and 0.5 < R0 < 2.5 the systemic velocity
ranges from 249 km s−1 < V
0<261 km s−1. Yet whatever systemic velocity we
choose to perform the transform, the model fitted to the transformed data by non-linear least squares is close to Equation 4.2.
V(R0) = 57.4 R
2.5
R2.5+ 1.02.5
4.2 For example, if we use a systemic velocity 260 km s−1 to transform the ve-
locity data to the disk plane, we find a best fit model with V0 = 61 km s−1,
η = 2.4 and R0 = 1.14. When we estimate the systemic velocity using this
4.3. RESULTS Systemic velocity kms−1 Frequency 0 20 40 60 80 100 190 210 230 250 270 290 310 170 190 210 230 250 270 290 310
(a) The distribution of systemic veloc- ity from each sample star. The dotted line shows the median systemic velocity
254 km s−1of the LMC. The standard de-
viation is 23 km s−1, the same as the he-
liocentric radial velocities.
Median values of resamples
Frequency 245 250 255 260 0 50 100 150 200 250
(b) Monte Carlo resampling to estimate error on systemic velocity, results in this distribution with median 254 and
95% confidence interval 248 km s−1 to
259 km s−1
Figure 4.18
iterating with systemic velocity 253 km s−1, and transforming the data to the
disk plane, we converge on the optimal model 4.2. With this model we arrive at the distribution of systemic velocities calculated on each star in Figure 4.18a.
The distribution of systemic velocity estimates is very close to a Normal distribution, with a slight low tail. The mean value is 253 km s−1, however
the median value, 254 km s−1 is a better estimator of the true value given the
slight non-Normality. The standard deviation of 23 km s−1reflects the standard
deviation of our sample of heliocentric radial velocities. Using a bootstrap Monte Carlo resampling method to estimate the range of possible values allows for the slight non-normality of our sample, and we obtain a 95% confidence interval for the 249 km s−1 to 259 km s−1 Figure 4.18b.
The carbon stars from Kunkel et al. (1997), which were used by van der Marel et al. (2002), form a ring around the periphery of the galaxy; as well as the curious central objects, which in the VDM model appear to be counter- rotating with a disk velocity of −30 km s−1. However with our model of the
inner galaxy counter-rotation disappears. If the systemic line of sight velocity is forced to 263 km s−1the model disk velocity at the centre is−20 km s−1. The
0.0 0.5 1.0 1.5 2.0 2.5 3.0 −150 −100 −50 0 50 100 150 200 Disk radius kpc Disk v elocity km s − 1
(a) Disk model fitted, plotted with the in- dividual disk plane velocity data points.
0.0 0.5 1.0 1.5 2.0 2.5 −150 −100 −50 0 50 100 150 200 R vd −150 −100 −50 0 50 100 150 200
(b) Spline fit directly to the in disk plane velocity data points has a similar global shape as the disk model fit.
Figure 4.19