# How does Hubble's constant affect the Earth's orbit

If Hubble's constant is $2.33 \times 10^{-18} \text{ s}^{-1}$ and the earth orbits the sun with average distance of 150 million kilometers; Does that mean the earth's orbital radius increases approximately $11\text{ m}/\text{year}$? Does the earth's angular momentum change? If so, where does the torque come from? If the angular momentum doesn't change, does the earth's orbital velocity (length of a year) change? If so, where does the lost kinetic energy go?

Aside: the 11 meters per year figure comes from Hubble expansion of space the distance of the earth's orbital radius integrated over an entire year.

$$(2.33 \times 10^{-18}\text{ s}^{-1}) (1.5 \times 10^{11} \text{ m}) (3.15 \times 10^7 \text{ s}/\text{year}) = 11 \text{ m}/\text{year}$$

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BTW-- You'll note that Henry and I have made use of the MathJax formatting utility that is active on the site---using LaTeX syntax to typeset mathematics. – dmckee Jul 22 '11 at 23:28

The reason the universe expands is gravitation, as described by Einstein's field equation. The evolution of the universe is governed by gravitation, as described by Einstein's field equation. Over cosmological scale, the universe can be seen as homogeneous and isotropic, with very small density of matter and radiation. The density of matter and radiation is too small to counteract the expansion, an effect of initial condition. In local areas, however, the density is many magnitudes higher, and the effect of expansion is all but counteracted by the binding gravitational attraction.

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Everything you've said is true, but it fails to answer the OP's question. One way to see that it doesn't answer it is that although you mention the Einstein field equation, everything you say is equally valid in a Newtonian expanding universe. The fractional rate of change in size due to cosmological expansion is 0 for a hydrogen atom, $\sim 10^{-41}\ \text{s}^{-1}$ for the earth-sun system (arxiv.org/abs/astro-ph/9803097v1 ), and $\sim H_o$ for a photon. I don't see how you would get that from "tends to confine." – Ben Crowell Mar 11 '12 at 1:44
I also wouldn't agree that "the reason the universe expands is gravitation." The reason it has been expanding, ever since the Big Bang, is inertia, and this would be just as true in a Newtonian model as in one based on GR. – Ben Crowell Mar 11 '12 at 1:49
@BenCrowell: To your second comment: By inertial the expansion will slow down, whereas in fact the expansion is speeding up, currently modeled by a non-zero cosmological constant, an effect of gravitation. – Siyuan Ren Mar 11 '12 at 1:58
@BenCrowell: To your first comment: I read your paper, and all I see is that according to the authors themselves, this topic, the effect of cosmological expansion on local systems, is highly contentious. I doubt your paper has settled the problem and become the consensus. – Siyuan Ren Mar 11 '12 at 2:04
@BenCrowell: Regardless, it is well known that gravitation bounds the solar system. Even cosmological expansion has an effect, it is infinitesimal small. I don't see how that invalidates my phrase "tends to confine" at local scale. – Siyuan Ren Mar 11 '12 at 2:08

For objects smaller than cosmic scale, such as atoms, planets and solar systems, the electromagnetic and gravitational forces that hold them together are not changing (as far as we know) and so those objects do not change size.

Between galaxies, so widely separated, there's just gravity, and that tends to average out due to every galaxy being surrounded by other galaxies in all directions. On a cosmic scale, galaxies are like a gas, with galaxies being the "molecules" and described by the idea gas equation. To account for gravity and finite size of the galaxies, we might use the Van der Waals equation or some other variation, but that's beside the point, useful only for increasing accuracy.

Hubble's constant describes the rate at which the "container" of the galactic gas is expanding, the way the density of galaxies decreases over time. In an ordinary gas such as air, when in an expanding chamber, certainly the molecules are not expanding. Likewise, neither are the galaxies changing their sizes, at least not for Hubble-related reasons.

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You're oversimplifying by treating atoms and solar systems as being the same. GR does predict that solar systems will expand, just not by very much: arxiv.org/abs/astro-ph/9803097v1 The size of a hydrogen atom is set by fundamental constants. The size of a solar system is not. – Ben Crowell Jul 25 '11 at 0:39

No. Hubble's constant roughly says how the distance between two objects at rest with the universe grows. It does not say that the distant between everything is growing - the size of the hydrogen atom is not increasing. (My size is increasing, but from dietary rather than cosmological sources.) The size of objects and orbits are maintained by a balance of forces (classically). To whatever extent one can think of the expansion of the universe as pushing the Earth and Sun apart, it is already taken into account in setting the Earth's orbit.