# Exploring Stellar Rotation Velocities and Dark Matter Insights

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## Chapter 1: Understanding Stellar Rotation

The preceding article, "Testing a New Equation for Gravity," introduced a formula aimed at estimating the rotational velocities of stars. This framework, known as the Kolmogorov equation, illustrates the inverse correlation between a star's velocity and its distance from the gravitational center. Essentially, it serves as a theoretical foundation for Newton's Inverse Square Law. Notably, this theory indicates that the historical context of a star—encompassing its age, temperature, and its movement relative to the galaxy's center—should be integrated into the velocity prediction equations.

The Kolmogorov equation accounts for minute temporal changes in stellar velocities. It also incorporates the prediction error for stars located nearer to the galaxy's center.

### Section 1.1: Challenges in Astrophysical Data Access

The lack of qualifications in astrophysics and limited access to advanced computational resources have hindered the availability of star data in galaxies. The analysis detailed in this article utilizes an Excel spreadsheet on a personal computer to conduct its investigations. Its primary goal is to furnish empirical support for the hypotheses presented in previous articles.

The SPARC database, now accessible online, contains data on 3,660 stars across 175 galaxies. The dataset utilized in the Excel analysis includes:

- Estimated age of the galaxy.
- Velocities of the bulge, disk, and gaseous masses.
- Distances of these masses from the galaxy's center.
- The universe's expansion rate at 72 km/second/megaparsec.

The analysis focuses on stars situated 2 kiloparsecs (kpc) or more from the galaxy's center, drawing results from 2,665 stars across 161 galaxies.

#### Subsection 1.1.1: Age and Temperature Adjustments

The analysis revealed that the mass of gas in galaxies should be treated differently than the masses of stars in the bulge and disk. This aligns with the hypothesis that certain gas forms can act as information carriers. For instance, when hydrogen gas is excessively hot near the galaxy's center, it becomes ionized and unsuitable for this role. Conversely, hydrogen gas cools further out and is more capable of carrying information.

The Kolmogorov equation for predicting star velocity (excluding the gravitational constant) is expressed as follows:

v₂² ~ {(mᵇ + mᵈ + mᵍ) + [(mᵍ * k₁) * (1 + k₂)]} / d₂

where:

- v₂ = velocity of a star;
- mᵇ = mass of stars in the bulge within radius d₂;
- mᵈ = mass of stars in the disk within radius d₂;
- mᵍ = mass of gas within radius d₂;
- d₂ = distance of a star or gas from the galaxy's center;
- k₁ = adjustment factor based on the average age and temperature of stars;
- k₂ = adjustment factor reflecting the average increase in distance from the center due to the universe's expansion, valued at 0.074 per billion years.

When k₁ is set to 0, the Kolmogorov equation simplifies to Newton's inverse square law. Various k₁ values were determined by analyzing the mean error and associated standard deviation across all stars, leading to a best estimate of 2.7.

Stars within 2 kpc of the galaxy center were deemed to require an adjustment factor of 0, as the differences in predictions between the two formulas are negligible. Furthermore, as previously discussed, gas near the center is too hot to carry information, thus k₁ is assigned a value of 0 for these stars.

The individual value of k₁ for a star relies on its age and temperature, which are not readily available. The formula for k₁'s variation with age and temperature is:

([13.7/A]*[4/T])/([{d₁/d₂}*{4/T}]+1)²

where A is the star's age, T is its temperature, and d₂ = d₁ * (1 + [A * 0.074]).

Table 1 illustrates adjustment factors.

## Chapter 2: Reducing Prediction Errors

According to astrophysicists, once a star's core temperature reaches approximately 10 million °K, hydrogen fusion commences, releasing energy. The ultimate temperature of the star is significantly influenced by its mass. Starting from an average initial temperature of around 4,000 °K, massive stars can exceed temperatures of 10,000 °K, leading to a lower k₁ value.

The simulation results indicate that for galaxies with stars orbiting 2 kpc or farther from their centers, the root mean square error (RMSE) for predicting star velocities using Newton's inverse square law stands at -19%, with a standard deviation of 28%. In contrast, the Kolmogorov equation yields an RMSE of -17% and a standard deviation of 25%. This suggests that modifying the velocity prediction formula to account for hydrogen gas's ability to instantiate information contributes to a decrease in average prediction error, implying that dark matter could be linked to this informational mass.

The Kolmogorov equation functions similarly to a differential equation, accounting for small temporal changes. Over extended periods, adaptations to the equation may be necessary. As discussed in Article 1, regarding the dark matter enigma, one approach to modeling significant time changes is to factor in errors linked to the velocity of neighboring stars. The magnitude of the error comprises the cumulative discrepancies over time between predicted and actual velocities.

When the Newton and Kolmogorov equations incorporate neighboring star velocities, substantial reductions in mean prediction errors and standard deviations are achieved. In these computations, a slight modification to the Kolmogorov equation is applied, adjusting k₁ from 2.7 to 0.3. Values around 0.3 do not significantly influence the equation's predictive effectiveness.

The revised equations are:

Pᵣ = Nᵣ — (Nr-1 - Ar-1)

and

Pᵣ = Kᵣ — (Kr-1 - Ar-1)

where:

- Pᵣ = predicted velocity of a star at radius r;
- Nᵣ = velocity predicted solely using star masses (Newton's equation) at radius r;
- Kᵣ = velocity predicted using both star masses and information (Kolmogorov equation) at radius r;
- Ar-1 = measured velocity of a star at radius r-1 (the nearest star closer to the galaxy center).

The comparative statistics for 2,665 stars indicate that these equations accurately predict stellar velocities with minimal standard deviations. This is particularly relevant since standard errors for estimated velocities in the SPARC database could exceed 5%. Furthermore, predicted star velocities necessitate additional "adjustments" to accommodate age and temperature data that are unavailable in the STARC database. While predictions from the Kolmogorov equation do not statistically differ from those of Newton's equation, the Kolmogorov predictions offer a theoretical basis for including the errors linked to nearby star velocities, reinforcing the notion that stellar movement history should be acknowledged.

The concept that space's fabric may encode its historical information aligns with Loop Quantum Gravity's interpretation of gravity. Carlo Rovelli describes space as a spin network comprising nodes that signify its elementary components, with their connections illustrating proximity relations. The interplay between these spin networks creates spacetime, influenced by transformations represented through spinfoams, which document the evolution of the spin network.

### Section 2.1: Torsion's Role in Stellar Velocity Predictions

The Kolmogorov equation, adjusted for torsion, posits that space's fabric experiences twisting effects. As highlighted in Article 22, Dr. Nikolai Kozyrev's research suggests that time possesses an energy that imparts a spiraling motion to space, which could be the source of celestial rotation. Kozyrev's theory proposes that this spiraling motion correlates with the Golden Spiral of Sacred Geometry.

Article 24 delves into the notion of whether information constitutes matter, exploring how interstellar hydrogen gas comprises particles with varying spin states that may warp space's curvature over time.

According to the insights on ancient Indian wisdom, it was believed that the Sun's equinoctial points would take 24,000 years to complete a full circuit around the Zodiac. Modern calculations suggest a current precession rate of 50.1" yearly, equating to a full cycle of 25,920 years. However, evidence is lacking to confirm that this precession rate remains constant.

The average distance between stars in the Milky Way varies based on proximity to the galaxy center, estimated between 2.2 and 3.5 light-years, translating to roughly 1 parsec. The changes in distances between neighboring stars arise from both space's expansion and stellar rotation around the galaxy center.

In the STARC database, the average distance between stars beyond 1.99 kpc is approximately 1 kpc, or 1,000 stellar units. A 2.1% prediction error in the Kolmogorov equation may stem from neglecting torsion's influence on stellar velocity. For instance, a 1 stellar unit distance increase could correspond to a 0.002% rise in predicted velocity, while 1 kpc would elevate velocity predictions by 2%. When all Kolmogorov equation predictions are adjusted upward by 2% for each kpc, the mean prediction error reduces to -0.2%.

Dark matter does not factor into explaining this reduction in the mean prediction error. According to SPARC database insights, if dark matter were present, it would be quantifiable as the discrepancy between the necessary mass for the observed velocity and the cumulative measured masses of bulge, disk, and gas. For stars located more than 1.99 kpc from the galaxy center, the average dark matter presence would exceed baryonic matter by a factor of 1.27, with a standard deviation of 189%. However, this analysis reveals that 26% of stars exhibit negative dark matter, indicating some stars have velocities lower than predictions based on Newton's inverse square law. Furthermore, 18% of stars possess less dark matter than that associated with a neighboring star closer to the center, challenging the conventional understanding that dark matter should consistently increase with radius. This necessitates a reevaluation of dark matter's definition to align with the Kolmogorov equation, raising the question of why a new type of matter is considered necessary.

## Chapter 3: Galactic Collisions and Historical Information

As explored in a forthcoming article, the means by which a star's historical information is transmitted may be linked to hydrogen atom spin. For instance, the density of molecules with negative spin could affect space's curvature. Given that information has mass, the apparent dark matter connected to a star might be attributed to the spins of hydrogen atoms, which could "flip" based on the surrounding matter's requirements.

Astrophysicists propose that the majority of galaxies have undergone at least one collision with another galaxy, resulting in an accumulation of historical star movement data influenced by these encounters. Prediction errors in the adjusted Kolmogorov equations may serve as indicators of how space's shape transforms following such collisions.

Einstein's visualization of gravity likens space to a rubber sheet, which becomes distorted under the weight of mass. In contrast, the Kolmogorov equation suggests that space's fabric behaves more like a memory foam mattress, with each section retaining the memory of previous objects resting upon it.

## Conclusion: Reevaluating Dark Matter's Role

This empirical study posits that dark matter is entwined with information. The discrepancies between anticipated star velocities using Newton's inverse square law and observed velocities are not the sole rationale for dark matter's integration into the Lambda Cold Dark Matter (ΛCDM) model that explains cosmic evolution. As elaborated in Article 24, astrophysicists contend that dark matter shapes the structure and evolution of galaxies. The relationship between information and dark matter within galaxies merits further in-depth exploration by astrophysicists studying the universe's evolution.

The pivotal question raised by this article is:

Should we acknowledge information as possessing mass?