If I had two tennis balls, one meter apart, at what velocity are they moving away from each other due to Hubble flow? (Assume they are in a zero G environment, and nothing impedes their movement.)
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Can we talk about points in the intergalactic medium instead (i.e. between the galaxies and thus far from any gravitating matter), rather than two tennis balls? Because on such small scales, the Universe doesn't expand, instead being held together by gravity.
Anyway, the present-day Universe expands at a rate of $H_0 = 70\,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$. That means that if the two points are 1 Mpc apart, they recede at $70\,\mathrm{km}\,\mathrm{s}^{-1}$. If they are 0.5 Mpc apart, they recede at $35\,\mathrm{km}\,\mathrm{s}^{-1}$, and if they are 1 m apart, they recede at $$v_\mathrm{rec} = H_0 \, d = 70\,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1} \,\times \, \frac{1\,\mathrm{m}}{3.08\times10^{22}\,\mathrm{m}\,\mathrm{Mpc}^{-1}} = 2.2\times10^{-21}\,\mathrm{km}\,\mathrm{s}^{-1}.$$
You can compare this to the gravitational field from a tennis ball at the distance of $r = 1\,\mathrm{m}$. A tennis ball weighs $m_\mathrm{tb} = 58.5\,\mathrm{g}$, so the acceleration is $$g = \frac{G\,m_\mathrm{tb}}{r^2} = 3.9\times10^{-10}\,\mathrm{cm}\,\mathrm{s}^{-2},$$ which is small, but sufficient to accelerate the tennis balls to collide with each other in roughly $t \sim \sqrt{2r/g} \sim$ one week, with a terminal velocity of $3\times10^{-4}\,\mathrm{cm}\,\mathrm{s}^{-1}$, i.e. 12 orders of magnitude larger than the expansion of the Universe.
Oh… now I ended up talking about tennis balls anyway…
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Hubble Constant is about 70 km/sec per Megaparsec.
Given the Milky Way Galaxy is about 34K parsecs in diameter, that means the velocity between the edges of the galaxy due to Hubble Flow is about 2.38 km/sec.
With additional math, we can see two objects 1 meter apart are moving away from each other at about 2.26x10^-18 meters/sec. So after 10,000 years, they would be 10^-7 meters further away from each other.
Not sure of the math exactly, but it is clear Hubble flow is really small at understandable distances.
Jiminion
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