In the past several years kinematic data sets from Milky Way satellite galaxies have greatly improved, furthering the evidence that these systems are the most dark matter dominated objects known. This paper discusses a maximum likelihood formalism that extracts important quantities from these kinematic data sets, including the amplitude of a rotational signal, proper motions, and the mass distributions. Using a simple model for galaxy rotation it is shown that the expected error on the amplitude of a rotational signal is
Since their initial discovery [
Aaronson [
With the advent of high resolution, multiobject spectroscopy, the velocity samples from the brightest dSphs initially studied in [
Not only has the past several years seen an increase in the kinematic data sets for the brightest dSphs, the number of known Milky Way satellites has more than doubled due to the Sloan Digital Sky Survey (SDSS). As of the writing of this paper, the SDSS has discovered 14 new Galactic satellites [
Several kinematics studies on the ultra-faint population of SDSS satellites have been undertaken in the past several years [
With the above data sets now available, it is becoming increasingly necessary to develop better theoretical tools to interpret them. An important aspect of the theoretical modeling will necessarily require an interpretation of the kinematic data sets for the population of MW satellites; a detailed understanding of these kinematic data sets will be important not only for determining the mass distributions of each individual system, but for a global comparison to theories of Cold Dark Matter (CDM) [
The primary aim of this paper is to discuss a maximum likelihood formalism that is used for extracting important physical quantities from dSph kinematic data sets. Section
Information on the kinematic properties of dSphs is extracted from the line of sight velocities of their individual stars. This section introduces the likelihood used in the data analysis and projections for the errors attainable on several parameters using the likelihood.
The probability for a velocity data set,
The form of (
The distribution function in (
Equation (
There may also be rotational motion, in addition to the dominant contribution from random motions, present in the galaxy. Though rotation is intrinsic to the dynamics of the system and is not purely geometric as that described by (
With the addition of each of the terms in (
From the likelihood function defined in (
The Fisher matrix is constructed by differentiating the log of the likelihood function in (
In the second term in (
The projected errors obtained using (
The projected one-sigma error on the amplitude of the rotation parameter,
In addition to their interesting applications for understanding the rotation and proper motion of the dSphs, the calculations presented in this section are crucial for uncovering properties of underlying dark matter distributions. For example a strong gradient may reflect ongoing tidal disruption, which would clearly affect dark matter mass modeling, as is discussed in more detail in Section
This section discusses an application of the maximum likelihood formalism introduced in Section
Several dSphs have kinematics data sets large enough that statistically significant constraints may be placed on the parameters
To extract the rotation and proper motion signal, a simplified model is considered by assuming that the likelihood function is characterized by the six parameters introduced in Section
In order to determine the probability distributions of the parameters
Figure
Probability densities for the velocity components of Sculptor transverse to the line of sight. The components
Figure
The probability density for the amplitude of the rotational signal in Sculptor, using the model discussed in the text.
Figure
This section discusses the extension of the maximum likelihood analysis developed in Section
Under the assumption that a star cluster is spherically symmetric, the orbital distribution of the tracer particles is isotropic, that mass follows light, and the cluster is isolated from any external gravitational potential, the virial theorem provides a mass estimate of
Of course for dSphs it is not consistent to assume that these systems are isolated, since they are orbiting within the extended dark matter halo of the MW. For dSphs orbiting with the MW halo, the minimum mass estimate above is particularly useful, as it in turn provides a conservative estimate of the radius at which particles would be stripped due to the MW potential. As an example consider the case of Segue 1, which is an MW satellite with a stellar luminosity
At the next level of detail from the dynamical perspective, an estimate for the mass of the dSphs may be obtained by appealing to the sphericallysymmetric jeans equation, assuming that the gravitating mass of the system consists of both stars and dark matter. The analysis here closely follows the treatment given in the appendix of Strigari et al. [
The spherical jeans equation is
Given the above assumption for the velocity anisotropy of the stars and for the shape of the dark matter profile for the galaxy, the likelihood function can now schematically be written as
Before performing an example calculation using (
The example considered here uses the velocity data sample from Fornax of Walker et al. [
Given the base set of parameters in
Figure
Projections for the error on the log of the mass within a given physical radius for Fornax. The position and the errors on the stars from the Walker et al. [
The posterior probability density for the mass distribution of Fornax is now determined directly from the kinematic data, and compared to the projected error on the mass distribution as determined from Figure
Figure
The probability density for the mass of Fornax within 0.6 kpc (a) and the mass within its stellar tidal radius (b), defined here to be 3 kpc.
Results of the calculations for the mass distributions of the entire population of dSphs are presented in [
The likelihood formalism introduced above does not give any information regarding the optimal parameterization of the dark matter mass profile. For example referring to the calculation above, is the einasto profile with just two free parameters an acceptable description of the data? Given the parameterization of the dynamics via the spherical jeans equation, we can answer this question and determine how many parameters are required to describe the mass profile given the maximum likelihood formalism. Moreover, we can determine how the parameterization of the density profile depends on the given data sets. For example Segue 1, with only 24 measured line of sight velocities, may require a smaller set of parameters than does Fornax, which has
To specifically answer the question of how to determine the appropriate set of parameters in maximum likelihood theory one may appeal to the bayes evidence. For the purposes here the evidence,
As an illustration, four different dSphs that span a wide range in their respective number of velocities are considered: Segue 1, Sextans, Sculptor, and Fornax. These dSphs have 24, 424, 1352, and 2409 stars, respectively; for the latter three galaxies we consider only those stars that have a probability of
For the “Baseline” 3 parameter model, the following range of parameter space is integrated over for Fornax, Sculptor, and Sextans:
Three different models are compared to this Baseline 3 parameter model: (i) a model in which the parameter space for
Model (i) is thus described by four parameters, while model (ii) is described by five free parameters, and model (iii) is described by six free parameters. In Table
Parameter ranges and evidences for various models. The columns are for different dSphs, and the rows are for different models as described in the text. Each entry in the table gives the ratio of the evidence for the given model,
dSph | Segue 1 | Sextans | Sculptor | Fornax |
---|---|---|---|---|
No. of stars | 24 | 424 | 1352 | 2409 |
Exp | 3.4 | 3.3 | 3.3 | 3.6 |
1.6 | 0.9 | 0.8 | 1.3 | |
5.0 | 4.3 | 4.1 | 4.9 |
The results for the ratio of the bayes evidence for the various models, relative to
The results in Table
This paper has discussed the analysis of kinematic data from Milky Way dwarf spheroidals, with a primary motivation of (
When modeling the mass distribution of the dark matter halos of the dSphs, degeneracies between model parameters affect the determination of the total mass profiles, even in the context of the simplest spherical models. To shed light on these degeneracies, this paper has discussed a new criteria for model selection applied to the dSph kinematic data sets, taking a step towards determining how many parameters are needed to describe the mass distribution of spherical halos. For the four dSphs studied here, chosen because they have a wide range of available line of sight velocities, it is shown that, assuming CDM-motivated Einasto profiles for the dark matter halos, models with variable velocity anisotropy are slightly preferred relative to those with constant velocity anisotropy. Further, central slopes for the dark matter profile that are found in CDM simulations are not a unique description of the data sets; both more cuspy and less cuspy models are allowed for the central slope. This is primarily due to the degeneracy between the central dark matter slope with the central stellar profile and the velocity anisotropy distribution [
Future photometric and kinematic data sets promise to further pin down the mass distributions of the dSph dark matter halos. Upcoming data for the ultra-faint satellites will be particularly important and may be able to show whether any tidal effects are present in these galaxies. Further more, development of nonspherical distributions for both the light and dark matters should be considered given these data sets (for initial results along these lines see [
The author is grateful to Andrey Kravtsov, Matt Walker, and Beth Willman for discussion and comments on this paper. I additionally acknowledge the anonymous referee for constructive comments and critique that improved the content and presentation. The support for this work was provided by NASA through Hubble Fellowship Grant HF-01225.01 awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under Contract NAS 5-26555.