The paper "Dark energy and the structure of the Coma cluster of galaxies" by Chernin et al.

The paper "Dark energy and the structure of the Coma cluster of galaxies" by Chernin et al. explores the influence of dark energy on the structure and mass of the Coma cluster of galaxies. The authors approach the Coma cluster as a gravitationally bound system embedded in the dark energy background described by the ΛCDM cosmology. They aim to determine whether the dark energy density is significant enough to affect the structure of such a large cluster.

Key Concepts and Methods:

1. Three Characteristic Masses:

• Matter Mass (Mᴍ): The total mass of matter (dark matter and baryons) within a given radius.

• Dark Energy Effective Mass (Mᴅᴇ or mᵉᶠᶠ): The mass equivalent of the dark energy's gravitational effect within the same radius. It is negative due to dark energy's repulsive nature. It is negative due to dark energy's repulsive nature and is defined as the difference between the gravitating mass and the matter mass, calculated as:

Mᴅᴇ (or mᵉᶠᶠ) = Mɢ - Mᴍ

• Gravitating Mass (Mɢ): The total mass that includes the effect of dark energy, defined as

Mɢ = Mᴍ + Mᴅᴇ (or mᵉᶠᶠ).

2. Zero-Gravity Radius (Rᴢɢ):

• The radius where the gravitational pull due to matter is exactly balanced by the repulsive effect of dark energy.

• Calculated as 

Rᴢɢ = [Mᴍ/(8π/3)ρᴅᴇ]^1/3, 

where Mᴍ is the matter mass, and ρᴅᴇ is the density of dark energy.

3. Matter Density Profile:

• The authors propose a new density profile that accounts for dark energy effects, improving the fit to observational data from 1.4 Mpc to 14 Mpc.

Findings:

1. Impact of Dark Energy:

• At small radii (R ≤ 14 Mpc), dark energy has a negligible effect on the mass distribution, so

Mɢ ≃ Mᴍ. 

• In this context: Mɢ is the gravitational mass. Mᴍ is the matter mass.

• At larger radii (R ≥ 14 Mpc), dark energy significantly affects the structure, and its repulsive effect becomes comparable to or greater than the gravitational attraction of matter.

2. Mass Estimates:

• The upper limit for the total size of the Coma cluster is approximately 20 Mpc, beyond which the cluster cannot remain gravitationally bound due to the repulsive force of dark energy.

• The total matter mass within this radius is estimated to be Mᴍ ≲ 6.2 × 10¹⁵ M⊙.

3. Comparison with Traditional Profiles:

• The NFW (Navarro-Frenk-White) and Hernquist profiles are traditional models used to describe the matter density of clusters but do not adequately account for dark energy.

• The new profile proposed by the authors provides a better fit to observational data, especially at larger radii where dark energy effects are significant.

Conclusion:

The study concludes that dark energy significantly affects the structure of the Coma cluster at large distances from the centre. The proposed matter density profile, which incorporates the effects of dark energy, provides a more accurate representation of the cluster's mass distribution. The findings suggest that the Coma cluster's total matter mass is capped at around 6.2 × 10¹⁵ M⊙, and its size is limited to about 20 Mpc due to the influence of dark energy.

Implications:

• Large-Scale Structure: The results highlight the importance of considering dark energy when studying the structure and mass of galaxy clusters.

• Observational Verification: Future observations can test these predictions, particularly the proposed upper limits on the Coma cluster's size and mass.

• Cosmological Models: The findings reinforce the ΛCDM model and its implications for the role of dark energy in the Universe.

• By integrating dark energy into the analysis, this paper provides a more comprehensive understanding of the Coma cluster's structure, offering insights into the broader effects of dark energy on cosmic structures.

Reference: 

Chernin, A. D., Bisnovatyi-Kogan, G. S.,Teerikorpi, P., Valtonen, M. J., Byrd, G.G., & Merafina, M. (2013). Dark energyand the structure of the Coma cluster ofgalaxies. Astronomy & Astrophysics, 553,A101. https://doi.org/10.1051/0004-6361/201220781 

Mass and Effective Mass: Matter, Gravitating Mass, and Dark Energy Impacts:

Soumendra Nath Thakur

ORCiD: 0000-0003-1871-7803

07-06-2024

Abstract:

This study serves as a supplementary resource for researchers investigating length deformation in classical and relativistic mechanics. It complements existing studies such as the 'Comparative Analysis of Length Deformation in Classical and Relativistic Mechanics' series, 'Dynamics between Classical Mechanics and Relativistic Insights', and 'Advancing Understanding of External Forces and Frequency Distortion: Part -1.'

The focus of this resource is on the intricate dynamics of mass and effective mass, exploring fundamental concepts like matter mass and gravitating mass. The relationship between effective mass and forms of energy is examined in detail, elucidating how effective mass accounts for resistance to acceleration or changes in velocity under gravitational forces, akin to the gravitational potential energy of inertial mass.

Through a detailed analysis and illustrative examples, including the study 'Dark Energy and the Structure of the Coma Cluster of Galaxies,' the interplay between matter and dark energy is deciphered. This offers profound insights into the complex dynamics of the universe, shedding light on the interaction between matter and dark energy.

This resource serves as a supplementary guide for researchers investigating length deformation in classical and relativistic mechanics. It complements studies such as the 'Comparative Analysis of Length Deformation in Classical and Relativistic Mechanics' series,[₁,¹, ₂,²] 'Dynamics between Classical Mechanics and Relativistic Insights'[₃,³], and 'Advancing Understanding of External Forces and Frequency Distortion: Part -1.'[₄,⁴] Delve into nuanced discussions on mass and effective mass, covering fundamental concepts like matter mass and gravitating mass. Explore the intricate relationship between effective mass and forms of energy, understanding how it accounts for resistance to acceleration or changes in velocity under gravitational forces, akin to the gravitational potential energy of inertial mass. Through examples like the study 'Dark Energy and the Structure of the Coma Cluster of Galaxies,'[₅,⁵] witness how this resource deciphers the interplay between matter and dark energy, offering profound insights into their intricate dynamics.

Keywords: Matter Mass, Gravitating Mass, Effective mass, Dark energy.

The Concepts of Mass:

1. Matter Mass (Mᴍ):

The matter mass (Mᴍ) is a measure of the amount of matter in a substance or object. Anything that has volume and mass is classified as matter. Although mass itself cannot be seen, it is quantifiable and can be measured. The basic SI unit for mass is the kilogram (kg). The only entities that are not matter are forms of energy.

Mass vs. Weight: Mass is often confused with weight, but they measure different things. Mass measures the amount of matter in an object, while weight measures the force of gravity acting on that object. The force of gravity on an object depends on its mass and the strength of gravity. If the strength of gravity is held constant (as it is across the Earth), an object’s mass is directly proportional to its weight, meaning greater mass corresponds to greater weight.

Volume measures the amount of space that a substance or object occupies. The basic SI unit for volume is the cubic meter (m³), but smaller volumes may be measured in cubic centimeters (cm³), and liquids may be measured in liters (L) or milliliters (mL). The method used to measure the volume of matter depends on its state.

2. Gravitating Mass (Mɢ):

The gravitating mass (Mɢ) is the mass of an object as measured in a gravitational field. This requires a gravitational field, a scale, and various known masses.

Inertial Mass vs. Gravitational Mass: Inertial mass and gravitational mass of an object are identical in value but differ in their measurement methods. Inertial mass is measured by assessing an object's resistance to changes in velocity, while gravitational mass describes the force on an object within a gravitational field. Inertial mass is derived from the concept of inertia—the tendency of objects to remain motionless or in uniform motion unless acted upon by a force. Gravitational mass is measured using a scale.

Since the motion of an object on a spring involves constant changes in velocity, and the behavior of the spring system is related to the mass on the spring, inertial mass is often measured by attaching a mass to a spring. This involves using a spring with a known spring constant (k), which describes the 'stiffness' of the spring.

3. Effective Mass (mᵉᶠᶠ):

Energy, while not matter, is equivalent to mass. The concept of effective mass (mᵉᶠᶠ) refers to the mass equivalent of the gravitational effect of forms of energy.

Effective mass accounts for the resistance to acceleration or changes in velocity of a particle when responding to a gravitational force. This is similar to how the gravitational potential energy of any inertial mass contributes to resistance to acceleration or changes in velocity.

In this context, the effective mass represents the mass equivalent of the gravitational effects of forms of energy, accounting for resistance to acceleration or changes in velocity. Thus, effective mass is the representation of the mass equivalent of energy's resistance to acceleration under the influence of gravitational force.

Therefore, the effective mass is the mass equivalent of the gravitational effect of forms of energy, representing their resistance to acceleration or changes in velocity when responding to specific forces. This concept is similar to the gravitational potential energy of any inertial mass, which also accounts for resistance to acceleration or changes in velocity.

Example:

The study "Dark Energy and the Structure of the Coma Cluster of Galaxies" by A. D. Chernin et al. explores the mass distribution within the Coma cluster, introducing the concept of gravitating mass (Mɢ), which represents the cluster's total mass, and effective mass (Mᴅᴇ or mᵉᶠᶠ), indicating the mass equivalent of dark energy's gravitational effect. The effective mass is defined as the difference between the gravitating mass and the matter mass (Mᴅᴇ = Mɢ − Mᴍ). This equation highlights the contribution of dark energy within a specific radius. By distinguishing between matter mass (Mᴍ), dark-energy effective mass (Mᴅᴇ <0), and gravitating mass (Mɢ = Mᴍ + Mᴅᴇ), the study provides a detailed understanding of the interaction between matter and dark energy in the cluster's structure and dynamics.

References:

1. Thakur, S. N. Comparative Analysis of Length Deformation in Classical and Relativistic Mechanics. Preprints.org. https://doi.org/10.20944/preprints202405.1271.v1
2. Thakur, S. N. Comparative Analysis of Length Deformation in Classical and Relativistic Mechanics: Part-2. Preprints.org. https://doi.org/10.20944/preprints202405.1332.v1
3. Thakur, S. N. Dynamics between Classical Mechanics and Relativistic Insights: ResearchGate. https://doi.org/10.13140/RG.2.2.21005.96481/1
4. Thakur, S. N. Advancing Understanding of External Forces and Frequency Distortion: Part -1. ResearchGate. https://doi.org/10.13140/RG.2.2.35236.28809
5. Chernin, A. D., Bisnovatyi-Kogan, G. S., Teerikorpi, P., Valtonen, M. J., Byrd, G. G., & Merafina, M. (2013). Dark energy and the structure of the Coma cluster of galaxies. Astronomy & Astrophysics, 553, A101. https://doi.org/10.1051/0004-6361/201220781

The Influence of Force Vectors on Resultant Vector:

Soumendra Nath Thakur

ORCiD: 0000-0003-1871-7803

05-06-2024

A force vector represents a force with both magnitude and direction, acting on a body in motion. The resultant vector (R) and its magnitude depend on the combined magnitudes of two force vectors.

Key Concepts:

Force Vectors:

• Force vector = Magnitude × Direction

• Gravitational force vector (g) acts vertically downward.

• Velocity vector (v) acts horizontally.

Resultant Vector:

• Resultant vector (R) is determined by combining g and v.

• R's magnitude depends on g and v.

Influence of v and g:

• When ∥v∥≫∥g∥, g becomes negligible.

• R is predominantly determined by v.

• R's magnitude can be approximated as ∥v∥.

Vertical Component g:

• In a scenario where v≫g, the vertical component g is insignificant.

• R is primarily influenced by v.

A force vector represents a force with both magnitude and direction, acting on a body in motion. The resultant vector (R) and its magnitude depend on the combined magnitudes of two force vectors.

When the magnitude of the velocity vector (v) exceeds that of the gravitational force vector (g), the gravitational force's influence becomes negligible. Consequently, the resultant vector (R) is predominantly determined by the horizontal velocity vector (v), leading to the effective velocity of the body being approximated as v, with the gravitational force having an insignificant impact on R.

In a scenario where a 10g body accelerates to a desired velocity, with v≫g, the resultant vector R is primarily determined by the horizontal velocity vector v, rendering the vertical component g negligible.

In this context, the vertical component (g) is insignificant, allowing the effective velocity to be approximated as v and rendering the gravitational force's influence insignificant

Description of the Perception of Effective Mass:

Soumendra Nath Thakur

ORCiD: 0000-0003-1871-7803

05-06-2024

Inertial mass under the influence of gravity relates to the perception of weight, as the weight of a given mass can vary depending on the distance between the centres of gravity of two massive bodies and their respective masses, assuming no external forces are acting on the bodies.

The perception of weight is generally based on the amount of mass a body has. In a fixed gravitational potential where the body is at rest, a heavier body will have more weight than a smaller one. Additionally, this perception of weight may feel heavier or lighter without any change in the amount of mass of the bodies involved when the distance between the centres of gravity of the bodies changes. This can occur when an external force is applied to one body relative to the other, resulting in either vertical separation or horizontal movement. The applied forces, whether vertical or horizontal, between the massive bodies lead to a perceived change in the weight of the moving body in any relative direction. The change is more significant in the vertical direction, but a higher speed in the horizontal direction can also cause a perceived change in weight when the horizontal motion component significantly affects the vertical component of gravitational attraction between the bodies.

This perception of changing weight corresponds to changes in the total energy of the moving body relative to the other. These changes in energy, whether potential or kinetic, cause a perceived change in weight, which is referred to as effective mass.

Application of Effective Mass through Newtonian Mechanics:

Soumendra Nath Thakur
ORCiD: 0000-0003-1871-7803

05-06-2024

Effective mass is a fundamental concept that arises from the equivalence of energy and mass, but it does not represent actual mass by mass equivalent energy that has intractability with the force. In various physical contexts, effective mass serves as a vital tool for understanding the behaviour of massive manifestation of energetic state under the influence of force or dynamics. It allows us to simplify complex systems and predict their behaviour accurately.

Effective mass manifests not only in the traditional sense of mass but also in the energetic state of particles. This manifestation is rooted in the fundamental relationship between mass and energy. Even in situations where mass is not converted into other forms of energy, the concept of effective mass remains applicable, aiding in describing the behaviour of particles and systems.

Whether in classical mechanics, nuclear physics, semiconductor physics, cosmology, or other fields, effective mass provides a valuable framework for understanding and predicting the behaviour of particles and systems. It bridges the gap between the mass-like manifestations of energy and traditional mass, offering insights into the dynamics of physical phenomena.

The concept of effective mass is crucial in understanding the gravitational dynamics of the Coma cluster. Effective mass incorporates both the traditional matter mass and the impact of dark energy, providing a comprehensive understanding of the cluster's gravitational behaviour. This research highlights the importance of considering dark energy in mass estimations of galaxy clusters, as it significantly influences their structure and dynamics.

The research on "Dark energy and the structure of the Coma cluster of galaxies" by employing Newtonian mechanics, describes the local dynamical effects of dark energy and defines three masses to characterize the structure of a regular cluster: the matter mass, the effective mass, and the gravitating mass. It reveals that the effective mass, which considers the influence of dark energy on the cluster's dynamics, is negative, indicating an antigravity effect.