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REVIEW 4 major objections 4 minor 59 references

Motion of Particles in Solar and Galactic Systems by Using Neumann Boundary Condition

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read One constant acceleration may explain dark matter and orbit shifts.

desk verdict A serious modified-gravity attempt whose solar-system test rests on an ad hoc scaling and whose galactic force law does not follow from the stated equation of motion by a factor of two. read the letter →

arxiv 1909.00673 v1 pith:AGVVNJGW submitted 2019-09-02 astro-ph.GA

Hossein Shenavar This is my paper · ORCID
classification astro-ph.GA
keywords NeumannboundaryconditionmodifiedequationofmotionconstantaccelerationperihelionprecessiongalacticrotationcurvesmassdiscrepancylowsurfacebrightnessgalaxiesNFWprofile
topics Dark Matter
open problems Dark Matter
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that a modified equation of motion with a constant inward acceleration $2c_1a_0$ added to Newtonian gravity governs dynamics from the solar system to galaxy scales. At galactic scales the extra term acts like a new force that flattens rotation curves and removes the need for dark matter; at solar scales it gives small perihelion precessions that fall inside observed residuals once the solar system is modeled as a uniform sphere. The paper derives closed-form rotation curves for exponential disks, shows that mass discrepancy grows near the acceleration $2c_1a_0$, and finds that a critical surface density $\sigma_0=a_0/G$ controls the curve shape. It also reproduces the small-radius $1/r$ form of NFW dark halos and their constant central surface density, and it reports generally acceptable fits for 39 low-surface-brightness galaxies.

What carries the argument

The load-bearing object is the modified equation of motion with the constant acceleration term, equivalent to adding a linear potential $2mc_1a_0r$ to the Newtonian potential. For a continuous mass distribution the new force from each element is proportional to the unit separation vector, producing the fourth-order Poisson equation $\nabla^4\Phi=4\pi G\nabla^2\rho_b-16\pi c_1a_0\rho_b/M$. For an exponential disk of scale length $R_d$, the circular velocity is $v^2(r)=4\pi\Sigma_0GR_d\,y^2[I_0(y)K_0(y)-I_1(y)K_1(y)]+8c_1a_0R_d\,y^2I_1(y)K_1(y)$, with $y=r/2R_d$; the ratio $\sigma_0/\Sigma_0$, where $\sigma_0=a_0/G$, decides whether the curve rises, stays flat, or declines. The small-radius behavior of the apparent halo follows from the asymptotic forms of the Bessel functions, giving $\rho_{DM}\approx(c_1\sigma_0/\pi)r^{-1}$.

What would settle it

High-precision ephemeris residuals for Uranus, Neptune, Pluto, and Eris would settle the solar-system claim: the paper's Table 1 predicts $a_0$-driven precessions of roughly 21, 41, 50, and 75 arcsec/cy respectively, far larger than the GR contributions (about 1.2, 0.6, 0.4, and 0.18 arcsec/cy). If the measured residuals are consistent with GR and exclude these values, Eq. (1) fails at solar scales; for galaxies, a stacked test of the paper's Eq. (30), $(v^2-v_N^2)/r = 4c_1a_0\,yI_1(y)K_1(y)$ across many rotation curves would also falsify the model if the proportionality does not hold.

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Extended reading notes

Core claim

The central claim is that Eq. (1), $\frac{d^2\vec{r}}{dt^2}+2c_1a_0\,\hat{e}_r=g(r)\,\hat{e}_r$, is the correct weak-field equation of motion at all scales, with $2c_1a_0\approx8.6\times10^{-11}\,\mathrm{m/s^2}$ acting as a fundamental constant acceleration. For a smooth mass distribution, this term becomes a linear potential and a force that, inside a body, scales as $r/R$; for an exponential disk it adds $8c_1a_0R_d\,y^2I_1(y)K_1(y)$ to $v^2(r)$, making rotation curves rise or stay flat in low-surface-density galaxies. The same term yields a mass discrepancy that is a universal function of centripetal acceleration, an apparent dark-matter density $\rho_{DM}\propto1/r$ with universal central surface density $c_1\sigma_0/\pi$, and solar-system perihelion shifts consistent with the observed residuals after the uniform-sphere correction.

Load-bearing premise

The load-bearing premise is that the solar system can be treated as a uniform sphere of radius 30 AU so that the constant acceleration on an inner planet is reduced by $r/A_{\mathrm{Neptune}}$; without this rescaling, the predicted Mercury precession would be about 4.7 arcsec/cy instead of 0.06, far above the roughly 0.14 arcsec/cy residual that observations allow.

Editorial extensions

If this is right

  • If Eq. (1) is correct, the galactic mass discrepancy is not dark matter: the same baryonic disk produces rising or flat rotation curves in low-surface-density galaxies without any extra halo.
  • The mass discrepancy of any spiral galaxy should collapse to a single universal curve when plotted against centripetal acceleration, becoming large only below about $2c_1a_0$.
  • Apparent dark halos must have density proportional to $1/r$ at small radii and central surface density $c_1\sigma_0/\pi$, independent of galaxy mass, matching the NFW shape and the observed constant core surface density.
  • Outer solar-system bodies should show large $a_0$-driven perihelion precessions: the paper's Table 1 lists about 21, 41, 50, and 75 arcsec/cy for Uranus, Neptune, Pluto, and Eris, so accurate ephemeris fits are a direct test.
  • In high-surface-density galaxies the Newtonian term dominates and the $a_0$ term is negligible; all galaxies should asymptotically approach the same centripetal acceleration $2c_1a_0$ at large radius.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the solar-system test is the sharpest falsifier because the paper's uniform-sphere correction is applied only to inner planets; without it Mercury would have about 4.7 arcsec/cy of $a_0$ precession, far above the observed residual, and if precision ephemerides show no large outer-planet precessions the model's solar-scale success would be an artifact of that correction.
  • Editorial inference: the paper fits $c_1$ as a free parameter per galaxy and reports a mean value about half the theoretical 0.065; a cleaner test would fix $c_1$ at the theoretically derived value and fit only masses, which would make the rotation-curve agreement either stronger or much weaker.
  • Editorial inference: if the apparent halo density is set by $c_1\sigma_0$ and is independent of baryonic surface density, stacked weak-lensing or satellite dynamics should show a universal enclosed-mass profile per unit stellar mass; this is a testable extension the paper does not perform.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The manuscript proposes that a modified equation of motion, d²r/dt² + 2c1a0 ê_r = g(r) ê_r, imported from the author's earlier Neumann-boundary-condition work, governs dynamics from solar-system to galactic scales. It derives a perihelion-precession formula, claims that inner-planet precessions are consistent with observations provided the solar system is modelled as a uniform sphere of radius 30 AU, develops a many-body force law, derives an exponential-disk rotation curve, and fits it to 39 LSB galaxies with c1 and M/L as free parameters. It also argues that the model explains the NFW 1/r inner profile, a constant central surface density for dark halos, and Renzo's rule. The central claim is that all these phenomena follow from Eq. (1) with a single new constant c1.

Significance. If the derivation were internally consistent, the paper would offer a parameter-economical, dark-matter-free explanation of galactic rotation curves and would make concrete falsifiable predictions, most notably Eq. (30). The paper has some genuine strengths: it obtains analytic Bessel-function expressions for the disk rotation curve, it uses public rotation-curve data, and it states explicit testable relations. However, the load-bearing derivation is not consistent: the many-body force law does not follow from Eq. (1), the solar-system agreement is obtained through an ad hoc rescaling, and the galactic fits re-fit the supposedly universal constant c1 separately for every galaxy. These issues prevent the paper from supporting its central claim, even though the question it addresses is interesting and the data are relevant.

major comments (4)
  1. [Section 3, Eq. (19)] The relative-motion equation derived from Eq. (15) contains an a0 term 4μc1a0, which for m<<M equals 4c1a0, twice the 2c1a0 appearing in Eq. (1). The equal-mass sanity check in the text is also incorrect: with m1=m2=m, μ=m/2, Eq. (19) gives (m/2)x'' = F_N + 2m c1 a0, not m x'' = F_N + 2m c1 a0. Thus Eq. (1) is not the test-particle limit of the many-body equation used for galaxies, and the statement that the same equation governs solar and galactic scales is not supported by the derivation.
  2. [Section 3, Eq. (21)] Eq. (21) does not follow from Eq. (20). Summing the pair contributions in Eq. (20) gives the second term mi * 2c1a0 ∫ ρ(x')(x'-x)/|x'-x| d³x', without the division by the total mass M that is introduced in Eq. (21). The appeal to Eq. (14) does not justify this normalization, since Eq. (14) is a relation between the forces on two particles and not a prescription for continuum integration. Because Eq. (25), the rotation curve used for all 39 galaxies, is derived from Eq. (21), the galactic predictions are not consequences of Eq. (1).
  3. [Section 2, Tables 1 and 2] The inner-planet agreement is achieved by modelling the solar system as a uniform sphere of radius 30 AU and multiplying the precessions by r/A_Neptune. Without this factor, Table 2 gives Mercury an a0 precession of about 4.7 arcsec/cy (0.06/0.0129), far exceeding the roughly 0.14 arcsec/cy residual allowed by the quoted GR value and its uncertainty. The uniform-sphere assumption is ad hoc rather than derived. The same correction is not applied to Table 1, where the predicted outer-planet precessions of 21 to 75 arcsec/cy are reported without any comparison to observations; taken at face value they would be ruled out by the existing solar-system bounds, since the GR values in the same table are all below 1.2 arcsec/cy.
  4. [Section 6, Table 3] The galactic fits re-fit c1 separately for every galaxy, with values in Table 3 spanning 0.013 to 0.13, roughly a factor of ten, and the paper itself notes that the mean c1≈0.03 is half the theoretical value 0.065. Since c1 is introduced as the theory's single new universal constant, the rotation-curve fits are essentially calibrations of the model to the data rather than tests of a prediction. No goodness-of-fit statistic is provided for Table 3, and the derived M/L values include a strongly discrepant case, F571-8 with M/L≈31, which the paper acknowledges cannot be explained by inclination. The claim that the fits are generally acceptable is therefore not quantitatively substantiated.
minor comments (4)
  1. [Throughout] The manuscript contains many typographical and grammatical errors, including 'begining', 'uncertanties', 'precission', and the reference in the text to 'Table 13' where Table 2 is meant; a careful editing pass is needed.
  2. [Section 4, figure captions] The captions of Figs. 2 and 3 use inconsistent notation for the density ratio (σ0/Σ0 in one and Σ0/σ0 in the other), which makes the parameter ranges in the figures hard to interpret.
  3. [Table 3] UGC5750 appears twice in Table 3 with different c1 values (0.013 and 0.014) and different source labels; this duplication should be resolved and explained.
  4. [Section 3, Eq. (23)] The fourth-order Poisson equation is stated with a reference to a paper 'Shenavar (2016 b)' that is listed as in preparation; since the present manuscript relies on that equation for its conceptual claims, the derivation should either be included here or the statement should be softened.

Circularity Check

2 steps flagged · score 6.0 of 10

Galactic rotation-curve fits reduce to fitting the model's own c1 parameter, which was itself calibrated on galaxy rotation curves in the author's prior work; Eq. (1) is imported by self-citation rather than independently derived.

  1. fitted input called prediction [Section 6 ('Data Analysis of 39 LSB galaxies'), Eq. (25), Table 3]
    "In this data analysis, the galactic mass of galaxies and the parameter c1 are the free parameters of the fit. Also we found that the best fitting results are obtained when we restrict c1 to be within the interval of 0.2×0.065≤ c1 ≤ 2.0×0.065 which seems to be reasonable due to our prior estimation of this parameter (Shenavar 2016 a). However, I should mention that the mean value that we found for c1 in this data analysis appears to be about half of the predicted one c1 = 0.065."

    The non-Newtonian term in the fitted rotation curve Eq. (25) is 8c1a0Rdy²I1(y)K1(y), so c1 sets the strength of the very correction being 'tested'. The paper explicitly makes c1 a free parameter of each galaxy fit and allows it to vary per galaxy (Table 3 gives c1 values from 0.013 to 0.13). Since the same c1 had already been estimated from the last data points of galactic rotation curves in Shenavar (2016a), the 39 LSB fits reduce to adjusting the model's own acceleration parameter to the data they claim to explain. Flatness at 2c1a0 is therefore imposed by the fitted amplitude, not predicted independently.

  2. self citation load bearing [Section 1 (Introduction) and Section 2, Eq. (1)]
    "We applied this equation to a sample containing 101 HSB and LSB galaxies and re-estimated the value of the Neumann constant which was found to be compatible with the prior evaluations from Friedmann and lensing equations. ... the modified equation of motion of this particle in a centrally directed gravitational field is as follows (Shenavar 2016 a): d²r/dt² + 2c1a0 êr = g(r) êr (1)"

    Eq. (1) is the single load-bearing dynamical law for every prediction in the paper, yet it is not derived here; it is imported by citation to the author's own prior paper. That prior paper not only introduced the equation but calibrated c1 using a sample containing 101 HSB and LSB galaxies, the same type of rotation-curve data that the present paper then uses to 'test' the model with c1 again free. The chain from Neumann boundary conditions to the fitted rotation curves is therefore carried by a self-citation whose empirical anchor is the very data class being fitted, rather than by an independent external theorem or measurement.

full rationale

The dominant circularity is in the galactic application: the non-Newtonian acceleration in Eq. (1) is controlled by c1, and Section 6 states that c1 is a free parameter of the fit, previously estimated from galaxy rotation curves in Shenavar (2016a). Consequently, the paper's headline validation, 'generally acceptable' rotation-curve fitting for 39 LSB galaxies, is a fit of the model's own strength parameter to the data rather than an independent prediction. The claimed NFW 1/r halo profile and constant central surface density are analytic consequences of the assumed linear potential with fitted c1, so they function as consistency checks rather than independent confirmations. The solar-system precession calculation is less circular: Eq. (13) is a genuine derived consequence of Eq. (1), and the inner-planet suppression via a uniform-sphere model with factor r/ANeptune is an ad hoc modeling choice rather than a fit; however, it inherits a c1 originally calibrated on galaxy data, so it is conditional on the same fitted parameter. The many-body derivation also contains an internal inconsistency, Eq. (19) giving 4μc1a0 for the binary while the continuum Eq. (21) reintroduces a 2c1a0/M form, but that is a derivation error rather than a circularity. Overall, because the central rotation-curve 'prediction' reduces in part to fitting c1, the score is 6.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

No new particles, fields, or forces are introduced; the 'dark matter density' in Eq. (31) is a derived effective quantity, not a new entity. The Neumann constant c1 is a new parameter but not an entity.

free parameters (3)
  • Neumann constant c1 per galaxy = 0.013 to 0.13, mean about 0.03; theoretical value 0.065
    In the rotation curve fit the paper states 'the galactic mass of galaxies and the parameter c1 are the free parameters of the fit.' Allowing the model's universal constant to vary per galaxy weakens the test.
  • Mass-to-light ratio M/L per galaxy = 0.2 to 30.9
    M/L is fitted for each galaxy with a lower bound of 0.2 imposed from population synthesis (Sanders 1996); many galaxies sit at the bound, so baryonic masses are not independently determined.
  • Scale length R_d for three galaxies = 0.65 times the tabulated value
    For F730-V1 and UGC 11648 (and attempted for ESO 2060140), the paper reduces the adopted disk scale by 35% post hoc to improve the fit (Fig. 9).
assumptions (4)
  • ad hoc to paper Modified equation of motion Eq. (1): d²r/dt² + 2c1a0 ê_r = g(r) ê_r
    The paper does not derive this from Neumann boundary conditions; it imports it from Shenavar (2016a). The entire paper's phenomenology rests on this unproved premise.
  • ad hoc to paper The N-body a0-force uses the same 2c1a0 acceleration for every pair
    Eq. (21) assumes a constant acceleration 2c1a0 between a particle and each mass element, yet the two-body relative equation (19) gives 4μc1a0. The factor-of-2 conflict is unresolved.
  • ad hoc to paper The solar system can be approximated as a uniform sphere of radius 30 AU to suppress inner-planet precession
    Section 2 multiplies inner-planet precessions by r/A_Neptune to make predictions fit; the same correction is not applied to the outer planets in Table 1.
  • domain assumption Galactic disks are infinitesimally thin exponential disks with no bulge
    Section 4 assumes surface density Σ=Σ0 exp(-r/R_d) and ignores vertical thickness and bulges; the paper acknowledges this affects inner parts.

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Cite this review

Pith. "Pith review of Motion of Particles in Solar and Galactic Systems by Using Neumann Boundary Condition." pith.science (2026). https://pith.science/paper/AGVVNJGW

@misc{pith2026190900673,
  author       = {Pith},
  title        = {Pith review of: Motion of Particles in Solar and Galactic Systems by Using Neumann Boundary Condition},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AGVVNJGW}},
  note         = {Machine review of arXiv:1909.00673}
}
abstract

A new equation of motion, which is derived previously by imposing Neumann boundary condition on cosmological perturbation equations (Shenavar 2016 a), is investigated. By studying the precession of perihelion, it is shown that the new equation of motion suggests a small, though detectable, correction in orbits of solar system objects. Then a system of particles is surveyed to have a better understanding of galactic structures. Also the general form of the force law is introduced by which the rotation curve and mass discrepancy of axisymmetric disks of stars are derived. In addition, it is suggested that the mass discrepancy as a function of centripetal acceleration becomes significant near a constant acceleration $ 2c_{1}a_{0} $ where $c_{1}$ is the Neumann constant and $ a_{0} = 6.59 \times 10^{-10} $ $m/s^{2}$ is a fundamental acceleration. Furthermore, it is shown that a critical surface density equal to $ \sigma_{0}=a_{0}/G $, in which G is the Newton gravitational constant, has a significant role in rotation curve and mass discrepancy plots. Also, the specific form of NFW mass density profile at small radii, $ \rho \propto 1/r $, is explained too. Finally, the present model will be tested by using a sample of 39 LSB galaxies for which we will show that the rotation curve fittings are generally acceptable. The derived mass to light ratios too are found within the plausible bound except for the galaxy F571-8.

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