We selected all the stream stellar particles within 3 kpc of the Sun from Robyn Sanderson's data set and applied the random errors expected for FGK dwarfs as estimated by Re Fiorentin et al. 2015.
Here, we present the resulting Lxy vs. Lz distribution of the “expected” local streams found using the “true” values (bottom panel), the final Gaia catalog (middle panel), and the current ground based surveys (top panel).
I use 4 stars randomly drawn from the tails of the simulated Pal 5 stream to fit for the halo parameters, with fixed bulge and disk parameters. This is a trivial test because I use the same potential to fit as the simulation was run in, I fix the disk and bulge, and I am using perfect data. Here is the marginal posterior over just the halo potential parameters (marginalizing over nuisance parameters I use to model the debris).
I didn't cut away all of the burn-in samples so there are some weird tails in the posterior…
Will try with observational uncertainties soon.
[Kohei2015] I use 50 randomly selected stars (without error) from the leading or trailing streams. I assumed the correct parameters for disc and bulge potential as well as the correct scale length for the halo potential. I model the orbital energy distribution in each tail as a Gaussian distribution and run an emcee code to derive the posterior distribution of .
The results for :
[Kohei2014] I selected in total 200 stars (without error) from the leading or trailing streams. I assume that the potential is flattened logarithmic potential and fit the data to derive the parameters . Although the trial potential is simplistic, the derived flattening of the (total) potential is in reasonable agreement with the flattening of the halo potential.
acceleration at Pal 5:
(for all 200 stars)
(for 100 stars in leading stream)
(for 100 stars in trailing stream)
I just fit a flattened logarithmic potential, so the PDF is for those two parameters (v_c and flattening q). I computed the acceleration at Pal 5 and get
a(Pal5) = 3.42 +/- 0.04 pc/Myr^2
The PDF shows 1, 2, and 3 sigma contours. The orbit fit shows different projections: I fit to the 200 blue points, the best-fitting orbit is shown in red.
Some details on the method: I fit 200 points without errors (but effectively assuming that the errors are smaller than the width of the stream, because I use an estimated width of the stream in comparing model vs. data) and the likelihood is just the best-fit orbit's likelihood for each trial potential (I don't marginalize over the orbit for each potential).
Orbit fits. Only trailing tail was used. “Realistic” errors have been assumed. ~ 1-2 km/s RV precision ~1% distance errors, 0.1-0.2 mas/yr. The potential was a Myamoto-Nagai disk + bulge + NFW halo with narrow Gaussian priors on everything except the halo.
The acceleration at Pal5 is 3.50 +/- 0.03 pc/Myr^2
I attempted to fit the potential with the location of the progenitor known and no errors. I used only 30 stars. I fitted both a logarithmic potential (just two parameters circular velocity Vc and flattening q) and the full potential used for the simulation with all parameters bar the halo parameters held fixed.
The blue histogram is the acceleration at Pal5 distribution using the logarithmic and green is using the correct potential form. The red line shows the true value.
4 VL2 streams have been observed from the Sun, with errors and fitted by orbits in the flattened NFW potential
Rotation curve recovery
I selected 200 stars from each of the streams 1-4 (no error is included). I assumed that the potential is flattened logarithmic potential and estimated .
The true parameters of the halo profile are total mass and scale radius kpc. The average galactocentric distance of stars in the fitting sample (Gaia Only) is 6.85 kpc; at this radius the total enclosed mass is . The maximum distance is ~20 kpc.
I have taken a closer look at the “Gaia Only 20 streams” suite using classical tools of kinematic studies, such as Toomre and Bottlinger diagrams. A Toomre diagram plots the orbital velocity (total velocity in the plane of the MW) against the velocity orthogonal to the galactic rotation and gives an indication of the “haloness” of an object (i.e. the deviation to a circular orbit in the plane). The Sun as a typical disk star lies near Theta=235 km/s and v_orth=16 km/s in this diagram.
Bottlinger W diagram (orbital velocity against W velocity) of 5 streams from the 20 stream suite. The intersection of the dashed lines indicates the location of the Sun in this diagram
Here is the 67% error contour on the parameters (the dashed lines are contours of constant enclosed mass; the red one is the enclosed mass at 6.85 kpc):
Here is a plot of the best-fit model in enclosed mass as a function of radius. The error bands are taken from the extremes of the contours in parameter space.
I selected every 1000th star from SphericalIsochrone.GaiaOnly.sub.txv.dq.txt (no errors) until I had gathered 100 stars, then fed the resulting (x,v) distribution (isochrone100.txt) to the Dirichlet process mixture method described in http://uk.arxiv.org/abs/1303.6099. That method was developed with smooth DFs in mind, but it has a side effect of preferring potentials that make the action-space distribution “sharp”.
No information about stream membership is used: the (new) method used to calculate sums over the most likely partitions of stars into clusters, but I haven't checked whether it correctly groups cluster stars with their siblings.
The marginal likelihood of the correct model (red blob) lies a factor below that of the peak.
Table of log-likliehoods dpm-stream.dat
I attempted to find the potential parameters using 100 members of a single stream (#41). I marginalized over all parameters using emcee. I used both error-free data and the data convolved with Gaia errors.
The red lines show the true values. The black dots and lines are for the error-free data. The green dots and lines are with errors. The blue lines show lines of constant acceleration at r = 4.2, 13, 20 kpc.
[Kohei2015] I use 50 randomly selected stars (without error) from a single stream. I model the orbital energy distribution in the stream (without distinguishing leading and trailing streams) as a Gaussian distribution and run an emcee code to derive the posterior distribution of .
The following plot shows the mGC3 pole count map for A. Font's gas-dynamical simulations, without errors.
The structure that corresponds to each of the detected pole overdensities is shown here in X-Y and X-Z projections.
After this, we assume every particle is a RC giant and simulate the errors using M. Romero's code, which has the latest post-launch Gaia error prescriptions (including the effect of stray light). The following left and right plots are respectively the mGC3 (3D+radial velocities+proper motions) and nGC3 (3D+proper motions) pole count maps
As can be seen in these plots, most of the substructure is not recovered, once we have added the errors. This is because streams in this simulation at located at very large distance in general (several tens of parsecs) and stars have very very large parallax errors.
Thus we simulate errors assuming we can have photometric distances for these stars, assuming 10% distance errors and GAia errors for the rest of the quantities, and we get the following pole count maps, again mGC3 on the left and nGC3 on the right:
As can be seen here, the nGC3 method works very well and is capable of recovering most of the substructure.
We're searching for tidal streams in Robyn Sanderson's data, which is a simulation of sort-of-MW-like satellites infalling in a spherical Halo. Nbody particles were assumed to be RC stars and errors were simulated using PyGaia.
We have worked on detecting tidal streams with Hao Tian's Rockstar method (reference here) and Cecilia Mateu's mGC3 method (Mateu et al. 2011,MNRAS,415,214).
Here's the original distribution of particles form the simulation including errors, color-coded by satellite ID.
This is the mGC3 pole count map on a log scale:
These are the stars associated to each of the poles identified with mGC3:
Next we will run mgc3 again having eliminated the stars we have detected as associated to the strongest 3 poles to enhance the peaks caused by the fainter satellites.
Did the same as with Robyn's streams.
Applied REWINDER with modified parameters. Because the particles in Hans his simulations are released at once and not from the Lagrange points, Adrian modified his code to match his model with the streams.
Using the axisymmetric logarithmic potential defined by two parameters and :
Here is a plot of the minimum misalignment between the frequency and angle structure of the stream as a function of these two potential parameters.
and here is the angle and frequency structure of the stream in the best-fitting potential defined as and
Constraining the stream with a axisymmetric logarithmic potential we obtain:
2. with N=4040 stars
Palomar 5 with simple potential:
I used the full three-component model for the MW and fixed all model parameters except the halo mass and flattening. I also fit the progenitor's distance, radial velocity, and proper motion. I selected 100 random points from the sample with no error to constrain the model. Although there is no added error, the stream does have some thickness in all phase-space coordinates, which introduce 5 extra parameters that were fit as well. The PDF of the halo mass and flattening looks like:
It is clear that the orbit-fitting algorithm fails in this case (which is expected). It underestimates the mass while overestimating the degree of flattening. Somewhat surprisingly, it seems to have recovered a model similar to the first attempt using the orbit-rewinder algorithm. In order to confirm this failure is purely due to the systematic biases of the orbit-fitting approximation, I produced a set of mock data generated from the orbit of a single particle in the generative potential. When I fit that set of I am able to recover the generative model:
These results clearly demonstrate how the orbit-fitting algorithm can fail for even small, cold streams. In terms of the physical quantities, I also made PDFs of the circular speed of the model at the location of Pal 5 and the Sun. Again, both of these are smaller than those of the generative model.
It is interesting to consider how the orbit-fitting algorithm fairs when errors in the data are included. To that end I began by selecting a subset of 100 random particles from the the Pal 5 data convolved with Gaia errors. However, many of the 'stars' in this data set are faint enough that Gaia will not be able to reliable measure their radial velocities. Thus, when the data is convolved with the expected errors using the PyGaia code, the resulting errors are >1000 km/s. Therefore, I limited my selection to particles with radial velocity errors that are less than 70 km/s. I then took the same subset of particles from the data sets that assume that using Spitzer spectra will give radial velocities of either 10 km/s or 1 km/s. The resulting data subsets are shown in the following figure, where the black, blue, and red points are the Gaia, 10 km/s, and 1 km/s error sets respectively.
Suing the same technique as earlier I fit each of the data sets. To be clear, the free parameters are the halo mass and flattening, the progenitor's distance, radial velocity, and proper motions, and five 'thickness' parameters for each of the phase space coordinates. The PDFs of the halo mass and flattening for each of the runs are shown in the figure to the right. In this figure the green, blue, and red contours are for the Gaia, 10 km/s, and 1 km/s errors respectively. The additional black contours are for a fit that does not include the radial velocities. Not only do none of the contours include the generative model, there appears to be a significant shift in the likelihood peak depending on the errors in the radial velocity.
To fully understand this evolution in the likelihood peak, I examined how the PDFs depend on each phase-space coordinate. To that end, I fit the 10 km/s data using the angular position plus the radial velocity, distance, and proper motion seperately, as well as a run not including the proper motions, and a run not including the radial velocity. In these fits, I give the model the same freedom as in the previous fits. The resulting PDFs are shown in the following figure. It is important to note that in this figure I show the full range of allowed halo masses and flattening. This figure is interesting as it shows that each of the phase-space coordinates of the data prefer slightly different models. This explains the evolution of the likelihood peak in the previous figure. As the error decreases, the 'weight' of the radial velocity data increases in the fit, thereby pulling the data towards it's preferred peak.
Another interesting facet of the previous figure is the sheer size of the PDFs for any of the angular position plus one other coordinate runs, especially when compared to the size of the PDFs where all the coordinates are used. The reason for the size evolution is only partly due to the increased number of data points. The much larger cause is the increased constraints on the progenitor. From the plot of the stream data itself, it is clear that many of the points are located roughly at the position of the progenitor. They provide powerful constraints on the initial position and velocity of the progenitor particle when all their phase space coordinates are included. To show this more clearly, I repeated the previous analysis, except this time I fixed the progenitor's position and velocity to those of the generative model. The resulting PDFs are shown in the following figure: Fixing the progenitor position has a profound effect on the PDFs of the runs using the angular position and radial velocity, distance, or proper motions alone. As more phase space coordinates are used in the fit, the effect of fixing the progenitor becomes much smaller. To be sure, the location of the peak itself shifts slightly, but the size of the PDF remains roughly constant when all phase space coordinates are used.
These results demonstrate two key things. Firstly, the effect of the progenitor's freedom, or lack thereof, is profound for the orbit-fitting technique. Secondly, as expected, the orbit-fitting algorithm fails for a single realistic stream. It seems that the biases in the model PDFs introduced by the orbit-fitting algorithm vary depending on the phase-space coordinates are examined.
Using a single stream does not match the mass profile particularly well. In the upper panel the mass curve of the generative model is shown in black and the fits using a variety of different individual streams are shown in color. The solid lines are the average mass and the shaded region encompasses the +/- one sigma levels. The lower panel shows the residuals for the fit.
Using multiple streams seems to steadily improve the fit to the generative mass profile.
An interesting question is how the parameterization affects the inferred mass profile. As a first step towards that I've fitted a logarithmic potential rather than the generative potential. In the following plot the black curve shows the generative mass profile, the red curve shows a best fit to the mass profile for a logarithmic potential and the blue curve shows my inferred mass profile. The best fit curve was calculating by minimizing the total residuals between the two mass curves. It is worth noting that even the best fit logarithmic model does a somewhat poor job of reproducing the generative model.