Dark Energy: Mass, Distance & Gravity Relationships

[chris2112]

Is there a function of mass that tells us the distance that two objects are no longer gravitationally bound and start moving away from each other due to dark energy? Is mass arbitrary at a certain distance? To make my question broader, can anyone show me equations describing relationships between the forces of gravity and dark energy on mass?

 

[Nabeshin]

Following http://arxiv.org/abs/astro-ph/0302506, we can define the rough length scale at which objects begin to disspiate due to dark energy by:
$$-\frac{4\pi}{3}\left(\rho + 3P\right)R^3 \sim M$$

Taking a normal value for $$\Lambda$$ and assuming an equation of state of -1, such that $$\rho = - P$$, I get

For a mass of $$1 M_{\odot}$$ , ~100 pc (Solar system),
For a mass of $$10^{10} M_{\odot}$$, ~ 200 kPc (Galaxy),
For a mass of $$10^{13} M_{\odot}$$, ~ 2MPc (Cluster).

This suggests that all three systems are gravitationally bound and will never be disrupted.