# Can mechanical angular momentum be transferred by gravity?

At PSE there are a few answers that explain that since the moon is responsible for tides and tides slow down spinning, L must be transferred to the moon.

This implies that if earth's L were reduced by an asteroid, this would not influence the moon's orbit.

Can you clarify:

• does the change of $\omega$ of the Earth alter the orbit of moon only when that change is attributable to the moon or in any case?
• would there be tides if there were no moon?
• what is the mechanism through which angular momenta can exchange magnitude? This is apparently impossible since there is no connection or, we we consider the obsolete Netwonian gravity, the connection is radial and never tangential?
• Given your question and answer at physics.SE, I suspect that the main problem here is a misconception based on how $F=GM_1M_2/r^2$ is taught. Strictly speaking, this only pertains to point masses. Newton found that this also pertains to non-point masses with a spherical mass distribution (i.e., Newton's shell theorem). It does not however pertain to objects with a non-spherical mass distribution. Newton himself knew this; he just didn't know how to solve the problem of arbitrarily shaped objects. The mathematical tools to address this weren't developed until ~100 years after Newton died. Jul 31, 2016 at 7:46
• Two objects with slightly different orbital radii around the Earth. The "higher" one moves slower. Connect them with a rigid rod - what happens? A torque is exerted. Jul 31, 2016 at 10:24
• $mv^2/r = GMm/r^2$, so $v =(GM/r)^{1/2}$, but $L = mvr = m(GMr)^{1/2}$. Jul 31, 2016 at 11:01
• I merely offer this as an example of how torques can be imparted gravitationally where rigid objects are of finite size or asymmetric. i.e. where they are not point masses and cannot be approximated as such. Jul 31, 2016 at 11:06
• Perhaps this diagram helps, but if not then I'm afraid I can't help either. goo.gl/images/A1F5Vl Jul 31, 2016 at 11:30

1. does the change of $$\omega$$ alter orbit of moon only when that change is attributable to the moon or in any case?

Angular momentum is a conserved quantity (more on this later). What this means is that if some object is undergoing a change in angular momentum, that angular momentum must necessarily be transferred to some other object or objects. The Earth is losing rotational angular momentum. It's rotation rate is slowing down. That means some mechanism must exist for transferring that angular momentum.

The Earth's angular momentum would change if the Earth was struck by an asteroid, but that would be a very, very small change. The loss of gases from the top of the atmosphere is also represents a negligibly small transfer of angular momentum, as does the interaction of the Earth's magnetic field with the solar wind. The Earth is far too large to be subject to the YORP effect, and The only mechanism left is gravitation. Almost all of the Earth's loss of rotational angular momentum due to a slowing rotation rate is via gravitation. A small amount is transferred to the Sun and to the Earth's orbit about the Sun. All that is left is the Moon. That vast majority of the Earth's lost angular momentum goes into making the Moon recede from the Sun.

Angular momentum is a conserved quantity. The conservation laws are very important in higher level physics (Lagrangian and Hamiltonian physics); they are assumed rather than derived. Even higher level physics derives the conservation laws from Noether's theorem. However, angular momentum can also be derived from the strong form of Newton's third law. Proving this is standard fare in every undergraduate and graduate classical mechanics text. For example, it's the subject of chapter 2 in Classical Dynamics Of Particles And Systems by Marion et al. and it's the subject of chapter 1 (starting on page 5) in Classical Mechanics by Goldstein et al. There are some nice online derivations as well, such as the week 1 lecture notes for a class on Central Forces at Oregon State University.

Another name for a force that obeys the strong form of Newton's third law is a central force. A central force is one in which the force

• Acts between pairs of particles that comprise a system of particles,

• Obeys the weak form of Newton's third law (equal but opposite forces),

• Is directed along or against the line connecting the two particles, and

• Is a function of the radial distance between the particles only.

Examples include Hookean spring forces, Coulomb forces, and Newtonian gravitation. The gravitational force between a pair of particles is given by $$\vec F = G M_1 M_2 / r^2 \hat r$$, where $$\hat r$$ is a unit vector pointing along the line that connects the two particles.

1. would there be tides if there were no moon?

Yes. The Sun also raises tides, but smaller in magnitude than those raised by the Moon. The heights of the tides raised by the Sun are a bit less than half those raised by the Moon.

1. what is the mechanism through which angular momenta can exchange magnitude? This is apparently impossible since there is no connection or, we we consider the obsolete Netwonian gravity, the connection is radial and never tangential?

There are two issues with this thesis. One is that Newtonian gravity is not an obsolete theory; it remains widely use in space exploration and in astronomy. A better way to look at it is that Newtonian gravity has a limited domain of applicability.

The other issue is the claim that Newtonian is purely radial. This is only true in the case of point masses or objects with a spherically symmetric mass distribution. It is not the case for objects with an aspherical mass distribution. For example, the Earth has a significant equatorial bulge due to its rotation. This equatorial bulge exerts a torque on satellites orbiting the Earth that causes the satellites' orbits to precess. Scientists take advantage of this. There's a special class of satellites, the sun synchronous satellites, whose orbits by design precess by 360 degrees over the course of a year.

With regard to the Moon, the rotation of the Earth makes the tides raised by effectively lead the Moon by a small amount. While this isn't Newton's tidal bulge, it has a similar effect on the Moon. Averaged out over the course of time, the waters on the near side of the Earth pile up slightly in front of the Moon while the waters on the far side of the Earth pile up slightly behind the Moon. Because those far side waters are further from the Moon, the near side waters dominate gravitationally, causing the Moon to accelerate.

• Comments are not for extended discussion; this conversation has been moved to chat. Aug 2, 2016 at 12:23