After learning a bit from the thread "The Dark Sky Ahead", I have tried to think through the dynamics of gravitational interactions in a simplified scenario. Unfortunately the GR math of the problem is over my head. My attempt below, using a simpler approach, leaves me quite confused.(adsbygoogle = window.adsbygoogle || []).push({});

If my analysis below is correct, this would seem to suggest that the approach that is commonly used, comparing escape velocity with Hubble velocity to determine if a system is bound with respect to the universe expansion, is wrong.

Consider an (almost) empty universe expanding exponentially with a single point mass and a test particle.

Let p be the test particle.

Let P be the point M.

Let M be the mass of P.

Let R be the distance between P and p.

Let h be the fixed constant value of the Hubble constant.

Let G be the Gravitational constant.

The questions I want to investigate are about stable configurations.

Scenario 1. p is stationary, that is, not moving with respect to P

Case (a): escape velocity = Hubble velocity

The square of the escape velocity

VThe square of the radial velocity of p relative to P due to the expanding universe_{esc}^{2}= 2 G M / R

VThe value of R for which these two velocities squared values are equal is_{exp}^{2}= h^{2}R^{2}

R_{1a}= (2 G M / h^{2})^{1/3}

Question 1a:

If p is undisturbed by any outside force, is this static configuration with R = R_{1a}possible. (I understand that if it is, any disturbance would cause p to eventually move to infinity.): gravitational acceleration = Hubble acceleration

Case (b)

The gravitational acceleration is

AThe Hubble acceleration is what p would experience relative to P if M = 0. It is_{g}= G M / R^{2}

A_{h}= h^{2}R

The value of R for which these two accelerations are equal is

RQuestion 1b:_{1b}= (G M / h^{2})^{1/3}

Same as question (1a) except that R = R_{1b}.

Note: Clearly Q1a and Q1b cannot both be answered "yes".

Scenario 2. p is in an orbit about P.

Question 2a: Is a circular orbit possible?

Question 2b: If so, is the orbital velocity affected by the universe expansion?

Question 2c: If so, how?

The following are my guesses at answers.

Guess (2a): yes.

Guess (2b): yes.

Guess (2c):

The square of the orbital velocity with h = 0 is

VThe square of the radial velocity of p relative to P due to the expanding universe is_{orb}^{2}= G M / R

VThe total velocity squared is the sum_{exp}^{2}= h^{2}R^{2}

VThe square of the escape velocity is_{tot}^{2}= G M / R + h^{2}R^{2}

VV_{esc}^{2}= 2 G M / R_{tot}= V_{esc}implies

G M / R = hand the value of R is^{2}R^{2},

R_{2c}= (G M / h^{2})^{1/3}

Note: R_{2c}= R_{1b}

My guess is that the inward gravitational acceleration must equal the sum of the outward accelerations: Hubble plus centrifugal.

G M / RWhen V^{2}= h^{2}R + V_{orb}^{2}/ R

V_{orb}^{2}= G M / R - h^{2}R^{2}_{orb}= 0,

G M / R = hThis is the same result as in Scenario 1 Case (b).^{2}R^{2}

The analysis in Scenario 2 seem to suggest the conclusion I included at the beginning of this post.

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# Orbit Dynamics in an Expanding Universe

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