Emergent Gravity Project

Reproducibility & Method

This page describes how the κ(r) profiles shown in the paper can be reconstructed directly from the SPARC dataset.

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1. Data source

The analysis is based on the SPARC database:

https://astroweb.cwru.edu/SPARC/

For each galaxy, SPARC provides radial profiles including:

  • Observed rotation velocity: Vobs(r)
  • Gas contribution: Vgas(r)
  • Disk contribution: Vdisk(r)
  • Bulge contribution (if present): Vbulge(r)

2. Constructing the baryonic velocity

The baryonic contribution is reconstructed as:

Vbar²(r) = Vgas²(r) + Υ_disk · Vdisk²(r) + Υ_bulge · Vbulge²(r)

where:

  • Υ_disk = mass-to-light ratio of the stellar disk
  • Υ_bulge = mass-to-light ratio of the bulge (if present)

Typical values used in SPARC analyses:

  • Υ_disk ≈ 0.5
  • Υ_bulge ≈ 0.7

The results are not highly sensitive to moderate variations in these parameters, but this dependence should be considered in detailed analyses.

3. Definition of κ(r)

The compressibility profile is defined as:

κ(r) = Vbar²(r) / Vobs²(r)

This quantity can be computed directly at each radial point provided in the SPARC data.

4. Practical computation steps

  1. Download SPARC rotation curve files
  2. Extract radial data for a given galaxy
  3. Compute Vbar² using gas, disk, and bulge components
  4. Compute κ(r) at each radius
  5. Plot κ(r) vs r

Optional:

  • Compute the gradient |dκ/dr|
  • Normalize for visualization

5. Interpretation

The paper identifies that κ(r) profiles do not follow a universal law. Instead, they appear to organize into a small number of distinct morphological classes.

The classification shown in the paper is qualitative and based on visual inspection of the reconstructed profiles.

A key observation is that these profiles appear structured rather than stochastic, suggesting that the dynamical discrepancy may depend on radial organization and structural properties rather than only on enclosed mass.

6. Notes on robustness

  • Results depend on the adopted mass-to-light ratios
  • Different galaxies show significantly different κ(r) behaviours
  • The classification is currently phenomenological

Further work may include statistical clustering or systematic parameter variation.

7. Robustness to mass-to-light ratio variations

The dependence of the inferred compressibility profiles κ(r) on the adopted stellar mass-to-light ratios was explicitly tested.

The analysis was repeated using the following parameter ranges:

  • Υdisk = 0.4 → 0.6
  • Υbulge = 0.6 → 0.8

Across this range, the qualitative structure of the κ(r) profiles remains largely stable. Only a small fraction of galaxies (~7–8%) change their classification under these variations.

This indicates that the observed class structure is not strongly sensitive to moderate changes in the baryonic mass modeling.

The classification remains primarily phenomenological, but the observed robustness suggests that the underlying patterns reflect genuine structural features rather than artifacts of parameter choice.