# Cosmic motions: The bigger the scale, the faster?

The rotation of the Earth around its own axis is at a speed of 465 meters per second at the equator. The revolution of the Earth around the Sun is at a speed of 30 kilometers per second. The revolution of the solar system around the galactic center is at a speed of 220 kilometers per second. And the Milky Way is moving towards the Great Attractor at a speed of 552 kilometers per second.

So it seems the bigger the cosmic scale of the motion, the faster it goes. Is there a law of nature that makes this inevitable? I do have some understanding of Kepler’s laws, especially the second that makes a statement about the speed of an orbital motion as a function of the size of the orbit; but to my knowledge these laws only apply to different bodies orbiting the same central body, so I don’t think they would make statements about a series of different motions such as those above.

Other than satisfying Newton’s Laws of Motion and Law of Universal Gravitation, I don’t think so.

Jupiter revolves around the Sun at around 13 km per second. It’s a more massive planet with a larger orbit than Earth and would seem to be traveling more slowly. So I don’t think your theory is always true.

That, I think, is a necessary consequence of Kepler’s second law; Jupiter orbits the same body as Earth but on a longer orbit, so that law requires it to go more slowly (the same way satellites in low Earth orbit move faster than satellites higher up). By “scale” I don’t mean two bodies orbiting the same body but on differently sized orbits; I mean moving up one level, in the sense that the previously orbited body now orbits yet another one.

I would say that broadly speaking, Newton’s laws would apply. As you jump up each “level”, you are dealing with a much more massive system with much larger orbits. So it would stand to reason that larger velocities are required to maintain orbital equilibrium. But I don’t know if that pattern has a specific name.

But note that these effects work in opposite directions, other things being equal. Greater mass means higher orbital speed, but a larger orbital distance means lower orbital speed.

The larger/denser the object the stronger the gravitational field (for orbits.) Different rotational speeds (“day lenghts”) is a somewhat different issue. The Earth’s day would be much shorter, for instance, without the influence of the moon, just as Mercury’s would be without the influence of the sun. And a star with a day lenght of weeks can collapse into a neutron star with a day lenght of microseconds.

For a simple two-body system orbital speed goes with the square root of M/R (mass over radius). I don’t know anything about the details of galactic orbits, but I think the same basic relationship must apply?

So when you’re consider a galaxy or multiple galaxies in a gravitationally-bound system, for a given density of stars the mass of the entire system will increase as the cube of the linear dimensions. So I think that implies that as scale increases, for a galaxy the effect of increasing mass will dominate over increasing orbital distance, and (other things being equal) orbital speeds will tend to increase with linear scale to the power (3/2).

Galaxies aren’t small enough to be treated as a point mass, which is what we do for the Sun when we compute orbits in the Solar System. Also there’s a lot more dark matter than there is baryonic matter. AFAIK, no one’s tried to compute the orbit of the Sun around the Galaxy, but we can say it’s not a simple ellipse.

In some sense, this will be true, because motion at the largest scales is governed by Hubble’s Law. But the systems you’re talking about are all smaller than that.

Details of orbits are not important because as long as you are dealing with a system of particles stably bound together by gravity, the average kinetic energy will equal -1/2 times the average potential energy.

This will give some relation between the mass distribution and the rotational velocities and velocity dispersion. I think your estimate completely ignores the density profile of the galaxy in question — compare the orbital velocities graphed here which definitely do not increase to the power 3/2.

From the article on velocity dispersion, “groups and clusters of galaxies have a wider range of velocity dispersions than smaller objects”, which may give a partial answer to the OP question as far as dispersion is concerned.

If you’re talking about circular or elliptical motions, centrifugal force becomes enormous for small distances and high speeds. The Earth would fly apart rotating with a surface speed of 220 kilometers per second.

But neutron stars are the size of cities, and some have surface rotational speeds as high as about 20% of light speed. Huge counterexample.