[Paper Review] Applying Liouville's Theorem to Gaia Data

Matthew R. Buckley, David W. Hogg+1 more|arXiv (Cornell University)|Jul 1, 2019

Stellar, planetary, and galactic studies1 references4 citations

TL;DR

This paper demonstrates a proof-of-concept method to reconstruct the original mass and density profile of tidally disrupted star clusters—like globular cluster M4—using Gaia data and Liouville’s theorem, which conserves phase-space density during adiabatic disruption. By minimizing phase-space entropy to correct for measurement and orbital errors, the authors recover the progenitor’s mass and King radius with reasonable accuracy, enabling new constraints on dark matter and galaxy formation.

ABSTRACT

The Milky Way is filled with the tidally-disrupted remnants of globular clusters and dwarf galaxies. Determining the properties of these objects -- in particular, initial masses and density profiles -- is relevant to both astronomy and dark matter physics. However, most direct measures of mass cannot be applied to tidal debris, as the systems of interest are no longer in equilibrium. Since phase-space density is conserved during adiabatic phase mixing, Liouville's theorem provides a connection between stellar kinematics as measured by observatories such as Gaia and the original mass of the disrupted system. Accurately recovering the phase-space density is complicated by uncertainties resulting from measurement errors and orbital integration, which both effectively inject entropy into the system, preferentially decreasing the measured density. In this paper, we demonstrate that these two issues can be overcome. First, we measure the phase-space density of the globular cluster M4 in Gaia data, and use Liouville's theorem to derive its mass. We then show that, for tidally disrupted systems, the orbital parameters and thus phase-space density can be inferred by minimizing the phase-space entropy of cold stellar streams. This work is therefore a proof of principle that true phase-space density can be measured and the original properties of the star cluster reconstructed in systems of astrophysical interest.

Motivation & Objective

• To develop a method for inferring the initial mass and structural parameters of tidally disrupted star clusters using only post-disruption kinematic data.
• To overcome the challenge of phase-space density degradation due to measurement and orbital errors in Gaia data.
• To demonstrate that Liouville’s theorem can be used to reconstruct the original phase-space distribution of a disrupted system, even when it is no longer in dynamical equilibrium.
• To show that minimizing phase-space entropy across different Galactic potentials enables accurate recovery of the progenitor’s original properties.
• To establish a foundation for using phase-space density as a conserved quantity to probe dark matter physics and Galactic assembly history.

Proposed method

• Apply Liouville’s theorem to conserve coarse-grained phase-space density during adiabatic tidal disruption, linking observed kinematics to the original mass of the progenitor system.
• Use Gaia DR2 data to measure the phase-space density of the globular cluster M4, focusing on stars with low measurement errors to preserve core overdensities.
• Minimize phase-space entropy by varying Galactic potential models to identify the orbit set that best reconstructs the original low-entropy distribution of stars.
• Map the reconstructed phase-space density back to the progenitor’s mass and King radius using the relation between phase-space volume and total mass.
• Utilize action-angle coordinates to better represent coherent stellar streams, improving phase-space density estimation in non-equilibrium systems.
• Account for systematic errors by treating measurement and orbital uncertainties as entropy injections, which are corrected via entropy minimization.

Experimental results

Research questions

• RQ1 Can Liouville’s theorem be used to recover the original mass of a tidally disrupted star cluster from Gaia data?
• RQ2 How can measurement and orbital errors—both of which increase phase-space entropy—be corrected to recover the true phase-space density?
• RQ3 Can phase-space entropy minimization across different Galactic potential models accurately reconstruct the progenitor’s original structural parameters?
• RQ4 To what extent can this method be applied to cold stellar streams that are no longer gravitationally bound?
• RQ5 Can this approach provide constraints on the initial dark matter distribution in disrupted systems, even when the systems are no longer bound?

Key findings

• The method successfully recovers the mass and King radius of the globular cluster M4 using Gaia DR2 data with reasonable accuracy, validating the approach as a proof of concept.
• Stars with low measurement errors preserve the core overdensity in phase space, enabling reliable reconstruction of the original phase-space distribution.
• Orbital errors and measurement uncertainties act as entropy injections that disperse phase-space density, but these effects can be mitigated by minimizing entropy across potential models.
• Minimizing phase-space entropy leads to the identification of the true Galactic potential and orbital paths that best reproduce the original low-entropy state of the disrupted system.
• The technique demonstrates that action-angle space is a viable coordinate system for measuring phase-space density in disrupted systems, improving robustness over position-velocity space.
• The approach opens a pathway to probing the inner dark matter profile of the Milky Way by recovering the original phase-space volume of disrupted substructures.