Mass–Property Relations (MPRs)

• MPRs are quantitative relations that connect mass variables to structural, dynamical, thermodynamic, and compositional observables.
• They typically adopt log-linear or power-law forms, enabling their use as empirical scaling laws and latent-mass inference tools in diverse systems.
• Applications range from galaxy and black hole relations to compact-star and planetary models, illustrating their broad scientific utility.

Mass–Property Relations (MPRs) are quantitative relations that connect a mass variable to one or more structural, dynamical, thermodynamic, or compositional observables. In the cited literature, the term is applied to several distinct constructions rather than a single formalism: empirical scaling relations such as galaxy and halo mass–radius laws, compact-star universal relations linking mass to $I$, $Q$, $\lambda$, and $\delta M$, nuclear mass relations built from local algebraic cancellations, and particle models in which additional “property” coordinates generate constrained mass spectra (Chiosi et al., 2019, Aranguren et al., 2024, Bentley et al., 7 Mar 2026, Delbourgo, 2012).

1. Conceptual scope

Across the surveyed literature, MPRs serve three main functions. First, they act as empirical scaling laws, typically fitted as power laws or log-linear relations, for systems such as galaxies, black holes, galaxy clusters, and planets. Second, they act as inference tools: a measured property such as $M_{\rm gas}$, $Y_{\rm SZ}$, $L_K$, $\bar\lambda$, or a neighborhood of nuclear masses is used to recover an unmeasured mass or a latent background mass. Third, they act as theory-level constraints, as in Delbourgo’s anticommuting scalar-coordinate framework, where generations and Higgs multiplets emerge from “properties” and the resulting mass spectrum depends on a small set of couplings (Mulroy et al., 2019, Farahi et al., 2017, Aranguren et al., 2024, Delbourgo, 2012).

Domain	Mass variable	Linked properties
Stellar systems, galaxies, clusters	$M_s$, $M_{\rm BH}$, $Q$0, $Q$1	$Q$2, $Q$3, $Q$4, $Q$5, $Q$6, $Q$7, $Q$8, $Q$9, $\lambda$0
Compact stars and planets	$\lambda$1, $\lambda$2, $\lambda$3	$\lambda$4, $\lambda$5, $\lambda$6, $\lambda$7, $\lambda$8, core and water mass fractions
Nuclei and particle models	$\lambda$9, $\delta M$0, fermion and Higgs masses	neighboring masses, mirror differences, Grassmann “property” coordinates $\delta M$1

A plausible implication is that “MPR” is best understood as a family resemblance term. What is common is not the observable set or the fitting machinery, but the use of mass as the organizing coordinate for high-dimensional structure.

2. Functional forms and statistical structure

The dominant phenomenology is log-linear. For cluster observables in LoCuSS, the mean relation is written as

$\delta M$2

while the BAHAMAS/MACSIS local-linear-regression formulation promotes both slope and intercept to running functions of mass,

$\delta M$3

This already encodes an important result: several halo MPRs are not globally self-similar, but exhibit mass-dependent slope and scatter (Mulroy et al., 2019, Farahi et al., 2017).

The residual model is frequently log-normal or multivariate Gaussian in log-space. Farahi et al. model $\delta M$4 as a multivariate log-normal, with covariance that varies with halo mass. In the hot-gas analysis of IllustrisTNG, TNG-Cluster, and FLAMINGO, $\delta M$5, $\delta M$6, and $\delta M$7 residuals are “very close to Gaussian in $\delta M$8,” $\delta M$9 is “nearly log-normal but with a small high-tail,” and $M_{\rm gas}$0 shows “strong positive skewness and heavy tails” (Farahi et al., 2017, Aljamal et al., 7 Jul 2025).

Compact-star MPRs take a different but still low-dimensional form. The extended universal relations are polynomial fits in $M_{\rm gas}$1,

$M_{\rm gas}$2

for $M_{\rm gas}$3. Planetary interiors use piecewise power laws of the form $M_{\rm gas}$4, with composition- and mass-dependent exponents (Aranguren et al., 2024, Skinner et al., 16 Apr 2026).

A recurrent technical theme is that the same observable can play very different roles: direct predictor, latent-variable constraint, regularizer, or proxy-quality benchmark. This suggests that the semantics of an MPR are inseparable from the inference problem for which it is calibrated.

3. Stellar systems, black holes, and chemical scaling laws

In the mass–radius plane, stellar systems from globular clusters to galaxy clusters occupy a structured locus rather than a single unbroken power law. The compiled ranges in the galaxy-cluster/early-type-galaxy study are $M_{\rm gas}$5–$M_{\rm gas}$6, $M_{\rm gas}$7–$M_{\rm gas}$8 for globular clusters; $M_{\rm gas}$9–$Y_{\rm SZ}$0, $Y_{\rm SZ}$1–$Y_{\rm SZ}$2 for dwarf galaxies; $Y_{\rm SZ}$3–$Y_{\rm SZ}$4, $Y_{\rm SZ}$5–$Y_{\rm SZ}$6 for early-type galaxies; and $Y_{\rm SZ}$7–$Y_{\rm SZ}$8, $Y_{\rm SZ}$9–$L_K$0 for galaxy groups and clusters. For ETGs with $L_K$1, reported fits include $L_K$2, $L_K$3 with $L_K$4 dex, and $L_K$5. The paper attributes the observed distribution to two combined evolutions: stellar virial/SFH effects and a halo-demography boundary set by the maximum halo mass $L_K$6, whose convolution with collapse loci produces a curved envelope and the “zone of avoidance” (Chiosi et al., 2019).

Black-hole MPRs appear in both classical two-variable and newer multi-variable forms. In the Aquila simulation of low-mass black holes, the $L_K$7–$L_K$8 relation at $L_K$9 is

$\bar\lambda$0

while the $\bar\lambda$1–$\bar\lambda$2 relation is

$\bar\lambda$3

The fitted redshift evolution is strong for $\bar\lambda$4–$\bar\lambda$5, $\bar\lambda$6 and $\bar\lambda$7, and weak for $\bar\lambda$8–$\bar\lambda$9, $M_s$0 and $M_s$1. The proposed interpretation is that $M_s$2–$M_s$3 tracks virial equilibrium, whereas $M_s$4–$M_s$5 reflects self-regulated black-hole accretion and star formation, with $M_s$6 from $M_s$7 to $M_s$8 (Zhu et al., 2012).

Symbolic regression substantially generalizes these black-hole scaling laws. Using 145 nearby galaxies with direct SMBH mass measurements and $M_s$9 host-galaxy quantities, the best linear 3-parameter relation is

$M_{\rm BH}$0

with $M_{\rm BH}$1, $M_{\rm BH}$2, $M_{\rm BH}$3, $M_{\rm BH}$4, intrinsic scatter $M_{\rm BH}$5 dex, and RMSE $M_{\rm BH}$6 dex. A relation including a pseudobulge flag reaches RMSE $M_{\rm BH}$7 dex, while the classical $M_{\rm BH}$8–$M_{\rm BH}$9 relation is quoted with $Q$00–$Q$01 dex. The AIC/BIC comparison similarly favors the 3-parameter relation, with AIC $Q$02, BIC $Q$03, versus AIC $Q$04, BIC $Q$05 for the classical $Q$06–$Q$07 law (Jin et al., 2023).

Metallicity provides another galaxy-scale MPR. The gas-phase MZR is represented locally by second-order polynomials such as

$Q$08

with $Q$09, or

$Q$10

with $Q$11. Stellar metallicity is likewise fit by quadratic or saturating forms. In the feedback-comparison study, mechanical feedback gives the best match to a number of observations up to redshift $Q$12, although the predicted gas-phase metallicities seem to be higher than observed at $Q$13 (Ibrahim et al., 2023).

4. Halo and cluster relations as mass proxies

For galaxy groups and clusters, MPRs are often evaluated by their mass-proxy quality rather than by formal tightness alone. In the generalized X-ray scaling-relation formalism, a three-observable relation

$Q$14

is restricted in the self-similar model by

$Q$15

Two projections were identified as especially efficient mass proxies: $Q$16 and

$Q$17

Applied to observational samples at $Q$18, these relations reduced the intrinsic scatter to a relative mass error below $Q$19 per cent, with typical relative error from the generalized scaling relation only of $Q$20–$Q$21 per cent on cluster scale and about $Q$22 per cent for galaxy groups. A specific caveat is that a calibration based only on relaxed systems can over-estimate hydrostatic masses in disturbed objects by about $Q$23 per cent (Ettori, 2013).

The LoCuSS analysis extends this logic by fitting a multivariate hierarchical Bayesian model to 41 clusters at $Q$24, simultaneously modeling weak-lensing mass, gas observables, stellar observables, the selection variable, and intrinsic covariance. It reports 30 scaling-relation parameters for 10 properties. All relations probing the intracluster gas are “slightly shallower than self-similar predictions,” and the stellar fraction decreases with mass. Among the observables analyzed, K-band luminosity has the lowest intrinsic scatter with a 95th percentile of $Q$25, while the lowest-scatter gas probe is gas mass with a fractional intrinsic scatter of $Q$26. No distinction is found between the core-excised X-ray or high-resolution SZ relations of clusters of different central entropy, whereas higher-entropy clusters have higher stellar fractions than lower-entropy counterparts with modest significance (Mulroy et al., 2019).

The BAHAMAS and MACSIS simulations emphasize the importance of local, mass-dependent calibration. Their local linear regression yields running slopes and scatters for $Q$27 and $Q$28 as functions of total halo mass. At $Q$29 and $Q$30, representative values are $Q$31, $Q$32, $Q$33, $Q$34 at $Q$35, evolving toward $Q$36, $Q$37, $Q$38, $Q$39 at $Q$40. The conditional likelihood $Q$41 is accurately described by a multivariate log-normal distribution, and the covariance is generally negative at fixed halo mass (Farahi et al., 2017).

The more recent hot-gas analysis of IllustrisTNG, TNG-Cluster, and FLAMINGO evaluates five observables: $Q$42, $Q$43, $Q$44, $Q$45, and $Q$46, for halos with $Q$47 at $Q$48. At $Q$49 and $Q$50, the reported intrinsic scatters are $Q$51 and $Q$52 for $Q$53, $Q$54 and $Q$55 for $Q$56, and much larger for $Q$57, namely $Q$58 and $Q$59 in IllustrisTNG and FLAMINGO respectively. The implied mass scatters at the same scale are approximately $Q$60. The conclusion is correspondingly sharp: $Q$61 and $Q$62 are the most robust mass proxies, whereas $Q$63 is the poorest mass proxy and displays strongly non-lognormal residuals (Aljamal et al., 7 Jul 2025).

5. Compact-star and planetary relations

In rotating neutron stars, the standard $Q$64–Love–$Q$65 framework is complicated by the distinction between the TOV background mass $Q$66 and the observable mass of the rotating star $Q$67. In the Hartle–Thorne expansion,

$Q$68

with

$Q$69

The extended $Q$70–Love–$Q$71–$Q$72 relations use $Q$73 to solve for $Q$74 rather than simply setting $Q$75. Over compactness $Q$76, $Q$77, and spin parameters $Q$78, the fits retain an EoS-induced scatter $Q$79 in $Q$80. For $Q$81, the maximum fractional inference errors are reduced from $Q$82, $Q$83, and $Q$84 to $Q$85, $Q$86, and $Q$87 for $Q$88, $Q$89, and $Q$90 in polytropes, and from $Q$91, $Q$92, and $Q$93 to $Q$94, $Q$95, and $Q$96 in MIT-bag models. The paper’s explicit point is that the common identification $Q$97 can be inconsistent (Aranguren et al., 2024).

Planetary MPRs are formulated as interior-structure relations between composition, thermodynamics, and radius at fixed mass. The validated 2026 low-to-intermediate-mass model includes a stellar-irradiated H/He envelope, a steam/condensed-water layer, a silicate mantle, and a two-component iron core, along with state-of-the-art EOS, non-adiabatic thermal structure, and melting. It reproduces Earth’s radius and moment-of-inertia coefficient to within $Q$98, Mars and the Moon’s to within $Q$99, and Mercury, Venus, and Europa’s to within $\lambda$00 or $\lambda$01. The model computes 32,971 planets across $\lambda$02–$\lambda$03, and for $\lambda$04 it fits piecewise power laws $\lambda$05. For Earth-like compositions the reported segments are $\lambda$06 for $\lambda$07–$\lambda$08, $\lambda$09 for $\lambda$10–$\lambda$11, and $\lambda$12 for $\lambda$13–$\lambda$14. A composition-dependent form is also given as

$\lambda$15

with $\lambda$16, $\lambda$17, $\lambda$18, and $\lambda$19. The authors report that radii are generally smaller than in literature mass-radius relations at low instellations and larger at high instellations, with the improvement comparable to observational uncertainties (Skinner et al., 16 Apr 2026).

These compact-star and planetary cases illustrate two different uses of MPRs: removal of systematic bias in latent-mass inference, and construction of high-fidelity forward models for composition retrieval.

6. Nuclear mass relations and particle-property mass generation

In nuclear-mass systematics, MPRs are often algebraic identities or near-identities on a local stencil. The combined Garvey–Kelson study begins from the 18 elementary GK six-neutron-six-proton configurations on a $\lambda$20 $\lambda$21 grid and constructs three optimized linear combinations. One relation is optimized to predict all small mass differences on the grid, one to predict the central mass, and one to predict the corner masses. On the AME 2020 $\lambda$22 set, the central nucleus can be determined with a $\lambda$23 keV standard deviation, any of the four corner masses with a $\lambda$24 keV standard deviation, and the per-difference metric achieves a $\lambda$25 keV standard deviation. A key correction to common usage is explicit: these Garvey–Kelson relations “do not broadly sum to zero as is sometimes assumed.” The same paper also discusses their use as evaluation metrics and regularization constraints in machine-learning mass models (Bentley et al., 7 Mar 2026).

The improved Kelson–Garvey relations for proton-rich nuclei use mirror-nucleus binding-energy differences $\lambda$26 and recursively incorporate more participating nuclei than the original KG or Isobar-Mirror schemes. On 31 measured proton-rich nuclei with $\lambda$27 and $\lambda$28, the root-mean-square deviation is $\lambda$29 MeV, compared with $\lambda$30 MeV for Kelson–Garvey and $\lambda$31 MeV for Isobar-Mirror relations. The paper then predicts the masses of 144 unknown proton-rich nuclei with $\lambda$32, and derives one- and two-proton separation energies $\lambda$33 and $\lambda$34 for studies of proton and diproton emission (Tian et al., 2013).

A different use of “mass–property relations” appears in Delbourgo’s framework of anticommuting Lorentz-scalar coordinates. One extends spacetime to

$\lambda$35

so that different monomials in $\lambda$36 carry distinct property quantum numbers and appear as different generations of four-dimensional fields. The theory has a single Yukawa coupling,

$\lambda$37

and a renormalizable quartic superHiggs potential,

$\lambda$38

In the two-coordinate case, the fermion masses are

$\lambda$39

with $\lambda$40 after the loop-motivated shift $\lambda$41. In the three-coordinate case, the fermion mass matrix in the $\lambda$42 basis yields two neutrino masses, two charged-lepton masses, and two “proton” masses, all determined by the single Yukawa $\lambda$43, the three superHiggs constants $\lambda$44, and the small loop parameter $\lambda$45. The five-coordinate case is identified as algebraically forbidding and likely to require symbolic-algebra automation (Delbourgo, 2012).

Taken together, these examples show that MPRs are not restricted to observational regression. They also include local cancellation identities, symmetry-based extrapolators, and Lagrangian constructions in which “property” variables constrain the admissible mass spectrum.

MPRs therefore form a methodological class rather than a single law. The cited work repeatedly shows that slope, scatter, covariance, and even the meaning of the mass variable can be scale-, redshift-, dynamical-state-, and task-dependent. This suggests that the main scientific value of an MPR lies less in its formal compactness than in whether its assumptions—self-similarity, log-normality, locality, virialization, smoothness, or background-mass identifications—remain valid in the regime where inference is attempted.