Soumendra Nath Thakur
ORCiD: 0000-0003-1871-7803
12-06-2024
Abstract:
This analysis examines the consistent use of the universal gravitational constant G in calculations pertaining to both the total gravitating mass (Mɢ = Mᴍ + Mᴅᴇ), encompassing dark matter and baryonic matter, and the effective mass of dark energy (Mᴅᴇ or mᵉᶠᶠ). Through equations derived within the Newtonian gravitational framework, the paper illustrates how the classical universal gravitational constant G is applied to understand gravitational effects within the context of dark energy. By employing the same fundamental constant throughout the analysis, the study ensures conformity with established gravitational laws, reaffirming the role of G in elucidating the dynamics of mass and energy in cosmological structures.
Keywords: Universal Gravitational Constant, Total Mass, Dark Energy, Gravitational Effects, Newtonian Framework,
The analysis of the research on the Coma cluster of galaxies considers the gravitational constant G as the fundamental constant used in both the effective mass of dark energy (Mᴅᴇ) and the total gravitating mass (Mɢ). The known universal gravitational constant G is indeed utilized for calculating the gravitational effects, including those due to dark energy.
Here's how G is applied in the context of the effective mass of dark energy and the total gravitating mass:
1. Effective Gravitating Density of Dark Energy:
The paper uses the equation: ρₑ𝒻𝒻 = ρ + 3P
For dark energy in the ΛCDM model, ρᴅᴇ is the density, and Pᴅᴇ = − ρᴅᴇ, leading to:
ρᴅᴇₑ𝒻𝒻 = ρᴅᴇ + 3Pᴅᴇ = - 2ρᴅᴇ < 0
This indicates that the effective density of dark energy is negative, which corresponds to an antigravitational effect.
2. Acceleration Due to Dark Energy:
The gravitational acceleration a(r) at a distance R from the centre of a mass Mᴍ within a uniform dark energy background is given by:
a(R) = - G(Mᴍ/R²) + (4ϖG/3)ρᴅᴇR = aɴ(R) + aᴇ(R)
Here, the second term represents the antigravitational effect of dark energy, and G is the universal gravitational constant.
aɴ(R) and aᴇ(R) are components of the radial acceleration experienced by a test particle due to gravity and dark energy, respectively.
• Newtonian Gravity Component aɴ(R):
This is the standard Newtonian gravitational acceleration due to a mass Mᴍ at a distance R:
aɴ(R) = - G(Mᴍ/R²)
Here:
• G is the universal gravitational constant.
• Mᴍ is the matter mass causing the gravitational attraction.
• R is the distance from the centre of the mass Mᴍ.
• Dark Energy Component aᴇ(R):
This is the acceleration due to the effect of dark energy, which acts as a repulsive force (antigravity) in this context:
aᴇ(R) = (4ϖG/3)ρᴅᴇR
Here:
• G is the universal gravitational constant.
• ρᴅᴇ is the density of dark energy.
• R is the distance from the centre of the cluster.
Combined Acceleration
The total radial acceleration a(R) experienced by a test particle at a distance R from the centre of a spherical mass Mᴍ in the presence of dark energy is the sum of these two components:
a(R) = aɴ(R) + aᴇ(R) = - G(Mᴍ/R²) + (4ϖG/3)ρᴅᴇR
In this equation:
• aɴ(R) represents the attractive gravitational force.
• aᴇ(R) represents the repulsive force due to dark energy.
The balance between these two forces determines the net effect on the particle's motion.
3. Zero-Gravity Radius (Rᴢɢ):
The zero-gravity radius Rᴢɢ, where gravitational and antigravitational forces balance each other, is derived using G:
Rᴢɢ = [Mᴍ/{(8ϖ/3)ρᴅᴇ}]⅓
This radius delineates the region where gravity dominates (inside Rᴢɢ) from the region where dark energy dominates (outside Rᴢɢ).
4. Dark Energy Mass (Mᴅᴇ):
The effective mass (mᵉᶠᶠ) of dark energy within a radius R is:
Mᴅᴇ(R) = (8ϖ/3)ρᴅᴇR³
This shows that Mᴅᴇ depends on ρᴅᴇ and R, but the gravitational effect of this mass is accounted for using G.
The calculations involving Mᴅᴇ and Mɢ are based on the Newtonian gravitational framework where the universal gravitational constant G is consistently used. The paper does not introduce a separate or modified gravitational constant for dark energy; instead, it applies the same G throughout the analysis, ensuring consistency with the established laws of gravity. This approach confirms that the known universal gravitational constant G is used for the effective mass of dark energy (Mᴅᴇ) as well as for other gravitational calculations in the study.
Reference:
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
