Chapter 2

Dispersion of Stellar Velocities.

While studying stellar communities, space researchers encountered another mystery, "the dispersion of stellar velocities in communities."

Researchers Vil'en (1977), Holmberg, Nordetrem and Anderson (2007), Aumer and Binney (2009) discovered that throughout the lives of stars, the dispersion of their velocities constantly increases, following a power law.

There is nothing mysterious about velocity dispersion in a star community. Not all stars in a community can move at the same velocity.

But there is a mystery in the dispersion of stellar velocities in a community. A regularity is observed in the dispersion of stellar velocities in a community. And this regularity has a power-law dependence.

With increasing age of stars in communities, the dispersion of their velocities increases. This fact indicates that the dispersion of stellar velocities in a community depends on stellar age. In a community of young stars, velocity dispersion is small. And velocity dispersion is large for old stars in communities. These facts indicate that not only does the velocity of stellar motion depend on age, but the dispersion of stellar velocities in communities also depends on stellar age.

How does the dispersion of stellar velocities depend on age? According to which law of physics does this dependence exist?

Above, in this section "Dark energy," in the first part, it was established that the physics of the "Dark Energy" effect lies in the energy properties of stars and obeys the law of physics – the law of conservation of momentum.

That is, changes in the velocity and kinetic parameters of stars obey the law of conservation of the star's momentum. In stars, changes in physical parameters influence changes in kinematic parameters.

Let's consider the influence of the law of conservation of stellar momentum on the dispersion of stellar velocities. With linear dependence, the value of velocity dispersion for stars with similar physical parameters should be the same throughout their lives. Actually, with increasing stellar age, velocity dispersion among them increases. Such increase indicates a power-law dependence of stellar velocity dispersions on their age.

Let's try to find this dependence.

Let's set up an equation for the dependence of stellar velocity dispersion on their age. To derive such an equation, it is necessary to introduce time measurement into the formula for stellar momentum.

In the formula for stellar momentum

                            I = M . V= const                                                               (2.1)

the mass parameter M changes over time, since part of the star's mass is emitted into cosmic space.

M₀ – star's mass at the initial moment, at its birth.
Mτ – star's mass after time τ since birth.

Mτ = M0 - ∆m

Where Δm – part of mass lost by the star through emission over time τ.

Δmi = Δm/τ – average value of mass lost by the star per unit time through emission.

Consequently, ∆m = ∆mi · τ, and Mτ = M₀ - ∆mi · τ. 

τ – time elapsed from the star's birth to the studied moment of its life, its age.
Let's write the formula for stellar momentum, taking into account its lifetime:

Iτ = Mτ . Vτ = (M0 - ∆mi . τ) . Vτ = const = I0 = M0 . V0

I₀ – star's momentum at the initial moment, at its birth.
Iτ – star's momentum after time τ since its birth.
V₀ – star's velocity at the initial moment, at its birth.
Vτ – star's velocity after time τ since its birth.

Formula for the star's velocity of motion, taking into account the law of conservation of momentum and its age:                      

∆vi  = ∆v/τ – average acceleration of the star's motion over time τ.
∆v = ∆vi . τ – increase in the star's velocity of motion over time τ.

After transformation, we obtain the formula for the star's velocity increase over time τ:

Formula for the star's average acceleration over time τ will be: 

As can be seen from formula (2.3), with increasing stellar age τ, its motion acceleration increases, which corresponds to research data. From formula (2.3), we can derive the formula for the star's mass loss per unit time through emission.

Knowing the mass, acceleration, and velocity of a star from formula (2.4), we can determine the mass the star emits into cosmic space per unit time. Of course, the result will not be precise, since for a precise result it is necessary to consider the density distribution of emission over the entire stellar surface. Also, it is necessary to consider energy emission from the star's "dark spots."

Stars born in communities often have similar physical parameters: mass, velocity of motion, age.

Let's set up a formula for velocity dispersion between two stars in a community:

 Where, ∆V1-2 – velocity dispersion between two stars in a community;
Mo1, Vo1 – initial mass and initial velocity of the first star;
Mτ1, Vτ1– mass and velocity of the first star after time τ1 since its birth;
∆mi1 – mass loss of the first star per unit time through emission into cosmic space;
τ1 – age of the first star;
Mo2 , Vo2 – initial mass and initial velocity of the second star;
Mτ2 , Vτ2 – mass and velocity of the second star after time τ2 since its birth;
∆mi2 – mass loss of the second star per unit time through emission into cosmic space;
τ2 – age of the second star.

To study the influence of the law of conservation of momentum on velocity dispersion in a star community, we can make assumptions and simplifications.

Variant #1.
Assume that, conditionally, two stars exist with identical initial parameters

Mo1 = Mo2 = Mo – initial mass of first and second stars;
Vo1 = Vo2 = Vo – initial velocity of first and second stars;
∆mi1 = ∆mi2 = ∆mi – mass loss of the first star per unit time through emission into cosmic space;
But, the age of the first star is greater than the age of the second star, τ1 > τ2 , ∆τ = τ1 - τ2 Let's set up the formula for velocity dispersion between first and second stars, considering the accepted simplifications and assumptions.

After transformation, formula (2.6) becomes:

Formula (2.7), velocity dispersion of two stars with identical initial physical parameters but different ages.

For analysis, it is necessary to decompose formula (2.7) into two components:

- numerator: Mo . Vo . ∆mi . (τ1 - τ2);

- denominator: (M0 - ∆mi . τ1) . (M0 - ∆mi . τ2).

In the numerator of formula (2.7), «Mo . Vo . ∆mi . (τ1 - τ2)», there is a product of constant numbers, not changing during the lives of the studied stars, considering our assumption ∆mi = const. And the value ∆τ = (τ1 - τ2) is constant, although throughout the lives of the two stars, values τ1 and τ2 increase, but mathematically expression (τ1 - τ2) is constant.

In formula (2.7), throughout the stars' lives, only the denominator changes
«(M0 - ∆mi . τ1) . (M0 - ∆mi . τ2)». Let's transform the denominator, substituting τ1= τ2 + ∆τ:
«(M0 - ∆mi . τ2)2 - ∆mi . ∆τ . (M0 - ∆mi . τ2)»
The value ∆τ = (τ1 - τ2) is constant and does not change with increasing ages of both stars. Formula (2.7) in this particular case becomes:

The value of the denominator decreases with increasing stellar age. The decrease in the denominator occurs quadratically with the age of these stars. Precisely this quadratic dependence of denominator decrease increases the velocity dispersion of stars with increasing age.

Variant #2.
Assume, conditionally, two stars exist with parameters:

Mo1 > Mo2 , Mo1 = Mo2 + ∆Mo,

Vo1 = Vo2 = Vo,

τ1 = τ2 = τ.

That is, two stars of the same age, with equal velocity, but different masses. We accept simplification: ∆mi1 = ∆mi2 = ∆mi .

Substitute the accepted simplifications and assumptions into formula (2.5):

Velocity dispersion of stars depends on the denominator of formula (2.7.3) and has a quadratic dependence on the age of these stars, as in the first variant.

In the second variant, inaccuracies are present:
For a star with larger mass M₀₁, momentum I01 is larger (I01> I02), mass loss ∆mi1 through emission is larger (∆mi1 > ∆mi2) than for a star with smaller mass. The value of stellar acceleration (∆vi) depends not only on the loss of part of its mass (∆mi) through emission but also on the star's own mass (Mo).
This variant is not correct and is presented to show the power-law dependence of stellar velocity dispersion on age.

Variant #3.
Assume, conditionally, two stars exist with parameters:

Mo1 = Mo2 = Mo

∆mi1 = ∆mi2 = ∆mi

τ1 = τ2 = τ

Vo1 > Vo2

Vo1 = Vo2 + ∆vo

Substitute the accepted assumptions into the formula for stellar velocity dispersion (2.5).

After transformation, we obtain the formula:

The numerator of formula (2.8) consists of constant parameters that do not change throughout the lives of the stars we study. The denominator of formula (2.8) decreases with increasing stellar age, which increases the velocity dispersion of these stars. Since

With increasing stellar age, this inequality increases, increasing the dispersion of stellar velocities. The dispersion of stellar velocities, with increasing age, increases and exceeds the dispersion values at their birth. In the third case, the formula for stellar velocity dispersion has a hidden power-law dependence.

In formula (2.8), lifetime τ is included in the coefficient of change of the already initial velocity dispersion of the stars.

From the considered variants of changes in stellar physical parameters, we can conclude:

- "Any difference in the physical parameters of stars (mass, velocity, age) leads to an increase in the velocity dispersion of stars in a community."

- The increase in velocity dispersion of stars with increasing age is proof of the influence of the law of conservation of momentum on the velocity (kinetic) characteristics of stars.

Conclusions:

Analytical study and calculations of stellar velocity dispersions in associations showed:

- The dispersion of stellar velocities increases with increasing age. The law of increase in stellar velocities and the law of increase in velocity dispersion between stars is embedded in the energy design of stars.

Changes in the dispersion of stellar velocities with increasing age are physically lawful and
related to the energetics of stars and the law of conservation of momentum.

- The acceleration of stellar motion and the increase in dispersion of stellar velocities with increasing age is proof of the influence of the law of conservation of momentum on the velocity (kinetic) characteristics of stars.
The acceleration of stellar motion in cosmic space occurs under the action of Newton's 2nd and 3rd laws, the law of conservation of momentum, and the law of conservation of energy.
That is, the physical effect of acceleration of stellar motion and acceleration of the expansion of the Universe, called "Dark Energy," is embedded in the energy design of a star. The acceleration of stellar motion and the acceleration of the Universe's expansion is a manifestation of the actions of the laws of physics: Newton's 2nd and 3rd laws, the law of conservation of momentum, and the law of conservation of energy.

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