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Re, or not Re: Developing R5≡ R−2 as a scale radius for galaxy sizes, masses, and mass-to-light ratios

  • Alister W. Graham
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Abstract

The effective half-light radius Re marks an arbitrary 50-per-cent light boundary, and scaling relations involving such `effective' radii and their associated surface brightnesses, μe, systematically (and undesirably) vary as the percentage changes. Here, the projected radius R5≡ R−2, where the logarithmic slope of the surface-brightness and intensity profile equals 5 mag/dex and -2, respectively, and where the luminosity per logarithmic radial interval is maximal, is developed as an alternative. It can be measured non-parametrically or with a parametrized fit. For the Sérsic R1/n family, R5=(2n/bn)nRe, with R5/Re->e1/6≈1.181 as n->∞. Reparameterizing the (now bn-free) R1/n model in terms of R5 and μ5 removes the non-linear Re-n coupling, and because the local slope is 5 mag/dex at R5, correlated measurement errors in R5 and μ5 better cancel when deriving the inferred total magnitude (∝μ5−5log R5). Additionally, an exact single-integral identity is provided to relate any projected light percentage to the percentage within a sphere of the same radius. The scale radius R5 is shown to be connected, through a weakly n-dependent factor to the anisotropy-insensitive internal radius r−3, yielding a refined n-dependent Wolf-type mass estimator M−3 and spatial mass-to-light ratio (Mdyn/L)−3. Specifically, M−3≈4G−1<σlos2>R−2≈4.72G−1<σlos2>Re. Past half-light substitutions in dynamical mass estimators introduce systematic Sérsic-dependent offsets of 12-18 per~cent in enclosed mass and offsets spanning >20 per~cent in the mass-to-light ratio within r−3. Dynamical mass estimators appropriate for (dark matter)-free stellar systems are also provided, and all relations are implemented in the companion Python package R5Tools.

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Cite

BibTeXRIS
BibTeX
@misc{graham2026developing,
  title = {{$R_\rm e$, or not $R_\rm e$: Developing $R_5\equiv R_-2$ as a scale radius for galaxy sizes, masses, and mass-to-light ratios}},
  author = {Alister W. Graham},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.17680},
  eprint = {2608.17680},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.17680},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - R_e, or not R_e: Developing R_5≡ R_−2 as a scale radius for galaxy sizes, masses, and mass-to-light ratios
AU  - Alister W. Graham
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2608.17680
UR  - https://arxiv.org/abs/2608.17680
AB  - The effective half-light radius R_e marks an arbitrary 50-per-cent light boundary, and scaling relations involving such `effective' radii and their associated surface brightnesses, μ_e, systematically (and undesirably) vary as the percentage changes. Here, the projected radius R_5≡ R_−2, where the logarithmic slope of the surface-brightness and intensity profile equals 5 mag/dex and -2, respectively, and where the luminosity per logarithmic radial interval is maximal, is developed as an alternative. It can be measured non-parametrically or with a parametrized fit. For the Sérsic R^1/n family, R_5=(2n/b_n)^nR_e, with R_5/R_e->e^1/6≈1.181 as n->∞. Reparameterizing the (now b_n-free) R^1/n model in terms of R_5 and μ_5 removes the non-linear R_e-n coupling, and because the local slope is 5 mag/dex at R_5, correlated measurement errors in R_5 and μ_5 better cancel when deriving the inferred total magnitude (∝μ_5−5log R_5). Additionally, an exact single-integral identity is provided to relate any projected light percentage to the percentage within a sphere of the same radius. The scale radius R_5 is shown to be connected, through a weakly n-dependent factor to the anisotropy-insensitive internal radius r_−3, yielding a refined n-dependent Wolf-type mass estimator M_−3 and spatial mass-to-light ratio (M_dyn/L)_−3. Specifically, M_−3≈4G^−1<σ_los^2>R_−2≈4.72G^−1<σ_los^2>R_e. Past half-light substitutions in dynamical mass estimators introduce systematic Sérsic-dependent offsets of 12-18 per~cent in enclosed mass and offsets spanning >20 per~cent in the mass-to-light ratio within r_−3. Dynamical mass estimators appropriate for (dark matter)-free stellar systems are also provided, and all relations are implemented in the companion Python package R5Tools.
ER  -