It is often said (often by Leif, especially) that the only mechanism of interest in changing solar spin due to barycentric motion would be tidal forces and that those are just way too small to matter. Yet there ARE other forces. Electromagnetic, for one. (As Adolfo is fond of pointing out, and occasionally posts interesting movies of homopolar motors – we know the sun dumps a lot of charged particles into space and it must do something. The question is how much…)
I had a tiny “Ah Hah!” moment when I realized that the guys managing satellites must have already dealt with the miscellaneous forces and might well have a language and an understanding that would be helpful. That lead to mention of electromagnetic forces in some of their literature, along with solar wind and light forces; but the one that caught my eye was gravity gradient torque. This is strong enough that it is regularly used in satellite attitude control (or must be overcome with heavy reaction mass / attitude control wheels).
Unfortunately, looking into it, the math is mind numbing and most all of it is centered around a very small thing orbiting a very large thing, not a very large thing orbiting a common center of mass. Furthermore, the math is specifically described as being not possible to solve in a closed way. There must be simplifying assumptions made, and those assumptions are orthogonal to the solar case. That means I can’t say “Eureka!”, only “Hmmmmm…..”
I’d started on this from the point of view that a simplified model of solar angular motion could be made (no, I didn’t finish it). The idea was just to say that the sun had most of its mass in the middle, so you could model point masses of 1, 2, 4, 2, 1 rotating about that central point with mass 4 when the barycenter was at the center, then look at the case where it moves to the edge. With L = r x mV
So at one point, the angular momentum would be a series of 1 x 2, 2 x 1, 4 x 0, 2 x 1, 1 x 2 when rotation was central and then 1 x 4, 2 x 3, 4 x 2, 2 x 1, 0 x 0 when the barycenter was at the edge. As those two cross product sums would be different, the question was “Where did the angular momentum come from / go to?” as it is a conserved quantity.
I figured that one could compute the AM of the whole sun vs the barycenter that way, then of the solar rotation compared to it’s center of mass, figure the change in one to allow for the change in the other, and look for any ‘order of magnitude’ similarity to what is actually seen in changed solar motion. (That ignores the change in the planetary motion, but if it’s at all close and of the proper sign, would support the notion of “Dig Here!” more precisely to find more evidence).
Before I even got to that point, I ran into The Falling Cat problem…
Basically, if you look at ALL the angular momentum problems / physics I was exposed to in school (and most of what folks talk about when discussing planetary / solar interaction) it is treated to the usual rules from that Angular Momentum wiki. Sliding past, hardly noticed, is “For a rigid body rotating around an axis of symmetry”. Right out the gate, we’ve violated those things. The physics we know and love does not apply. (Well, it sort of applies, but in a very messy way we can’t really solve well). Thus the falling cat problem.
How does a falling cat turn itself feet down while not violating conservation of angular momentum?
It depends on the fact that the cat is not a ‘rigid body’ and may not be rotating about its ‘axis of symmetry’. As the sun is not rigid, it will follow ‘different rules’ from what folks say when they trot out their high school physics and conservation of angular momentum rigid body solution mindset. Just like the cat. That is why so few folks can answer the falling cat question.
The cat, you see, bends in the middle. It acts like two different cylinders, with a pivot at the waist. That lets it combine two different spin actions into one overall rotation of the feet downward.
The falling cat problem consists of explaining the underlying physics behind the common observation of the cat righting reflex: how a free-falling cat can turn itself right-side-up as it falls, no matter which way up it was initially, without violating the law of conservation of angular momentum.
Although somewhat amusing, and trivial to pose, the solution of the problem is not as straightforward as its statement would suggest. The apparent contradiction with the law of conservation of angular momentum is resolved because the cat is not a rigid body, but instead is permitted to change its shape during the fall. The behavior of the cat is thus typical of the mechanics of deformable bodies.
The solution of the problem, originally due to (Kane & Scher 1969), models the cat as a pair of cylinders (the front and back halves of the cat) capable of changing their relative orientations. Montgomery (1993) later described the Kane–Scher model in terms of a connection in the configuration space that encapsulates the relative motions of the two parts of the cat permitted by the physics. Framed in this way, the dynamics of the falling cat problem is a prototypical example of a nonholonomic system (Batterman 2003), the study of which is among the central preoccupations of control theory. A solution of the falling cat problem is a curve in the configuration space that is horizontal with respect to the connection (that is, it is admissible by the physics) with prescribed initial and final configurations. Finding an optimal solution is an example of optimal motion planning (Arbyan & Tsai 1998; Xin-sheng & Li-qun 2007).
In the language of physics, Montgomery’s connection is a certain Yang-Mills field on the configuration space, and is a special case of a more general approach to the dynamics of deformable bodies as represented by gauge fields (Montgomery 1993; Batterman 2003), following the work of Shapere and Wilczek (Shapere and Wilczek 1987).
At that point, I realized I was not going to find a nice formula for momentum, apply it to the sun, and be done.
So that’s ONE body, solid but flexible. Without massive electrical and magnetic fluxes and without the occasional mass ejection. (Mine, at least, waits until on the ground and in the litter box before undergoing ‘mass ejection’ ;-)
It was just about there that I figure I’d not be able to solve this, given that the cat problem was only solved in 1969 and was still being expounded in 2003.
That led to the notion of satellite mechanics instead. Where I found that the “axis of symmetry” matters. Satellites are sometimes spin stabilized via the long axis, but can sometimes be spin stabilized on a minor axis (but only under specific conditions). What happens if the spin moment gets a bit off from the axis of symmetry? Well, the thing starts to ‘have issues’. There’s a lot of effort put into avoiding that case. From reaction mass, to control wheels, to using magnetic devices against the earth’s magnetic field.
“Right Quick” it sunk in that things in orbit are subject to a whole lot of forces that act in a variety of poorly understood ways to perturb them into various undesired spins, tumbles, and problem states. As these are often very rigid bodies, it is NOT just “tidal forces”…
Has an interesting discussion of some of the issues.
First things first: Suppose that a specific (constant) angular velocity will keep a satellite in the desired attitude indefinitely. If that angular velocity vector is aligned with one of the principal axes of the satellite and if the satellite has that exact angular velocity, then some external torque will be needed to make the satellite veer from the desired attitude. That second if is a mighty big if; it is impossible to make a satellite’s rotation rate be exactly as desired. The reason attitude control is needed is to counter internal errors in pointing and to counteract those external torques.
Most satellites do not use thrusters for attitude control, at least not as the primary means of attitude control. Many satellites do not even have attitude control thrusters. Many do not have any thrusters at all. The Hubble, for example, does not have any thrusters.
Many techniques are used to keep a satellite pointed in the right direction. Early satellites were spin stabilized. The whole satellite was set spinning either about its minor or major rotational axis. For a rigid body, rotation about either the major principal axis or minor principal axis (the principal axes with the largest and smallest moments of inertia) are stable. (This is not true for non-rigid bodies such as spacecraft with solar arrays. The only stable axis is the major axis.)
OK, first off we’re talking about a constant angular momentum. What happens when it changes? Then it has to be on a specific axis of rotation. What happens when it’s off axis? What happens when the thing is a ball of fluids? And we have the point that an orbiting body has external torques. And, once again, are reminded that bendy things have even more trouble being stabilized.
One approach is purely passive: Put the vehicle in an attitude where the perturbing environmental torques will put the vehicle back in that attitude should the vehicle drift from the desired attitude. The desired attitude for many satellites is nadir pointing. The satellite needs to be pointing toward the Earth and it needs to rotate at one revolution per orbit to maintain that pointing. The principal perturbing torques on a satellite in low Earth orbit are gravity gradient torque and torque from atmospheric drag. One way to take advantage of these environmental torques is to make the vehicle have a longish cylindrical shape with the cylinder axis pointing radially. Aero torque will be fairly small, and gravity gradient torque will naturally keep the vehicle in this vertical orientation.
OK, the sun is not a cylindrical shape and is not pointing toward the barycenter, I think this will matter. It implies to me that various angular momentum perturbations can cause various bits to wander around a ways from the expected places and this is likely to show up as various kinds of fluid flow, being as the sun is not a solid body. Those, then, are likely to stir the pot on magnetic and electrical forces that will even further perturb things. This is not looking like a ‘nothing happens’ scenario to me. But at least we don’t have any ‘aero’ forces (other than the solar wind).
But the ‘gravity gradient torque’ was what got my attention. Leading to:
Analytic signals of the gravity gradient tensor and
their application to Euler deconvolution
Department of Earth sciences, Uppsala University, Uppsala, Sweden
The analytic signal concept can be applied to the gravity gradient tensor data in three dimensions. For
the gravity gradient tensor, the horizontal and vertical derivatives of gravity vector components are
Hilbert transform pairs. Three analytic signal functions then are introduced for the x , y and z
components. The amplitude of the first vertical derivative of the analytic signals of x and y
components enhances edges of the causative bodies. It’s also shown that the directional analytic
signals are homogenous and they satisfy the Euler’s homogeneity equation. The application of the
directional analytic signals to the Euler deconvolution is demonstrated on some generic models to
locate causative bodies. The advantage of this method is that it allows the automatic identification of
the structural index from solving three Euler equations derived from the gravity gradient tensor for a
collection of data points in a window.
Oh dear. Tensors. My brain said “no”…
It is also shown that the
directional analytic signals are homogenous and satisfy the Euler’s homogeneity equation. The
advantage of this method is that the constant base level is removed from the Euler’s equation. Then the
structural index and source location can be estimated from Euler deconvolution of the directional
analytic signals directly. This improves the conventional Euler deconvolution using three gradients of
the vertical component of the gravity vector.
The key takaway for me, here, was just that this is NOT a tide force (or perhaps a tide force but without a tide), and it is not easy to calculate, even for a small rigid body.
6.1 Environmental Torques
Important environmental torques affecting satellite attitude dynamics include gravity
gradient, magnetic, aerodynamic, and solar radiation pressure torques. We develop
expressions for these torques in this section, and in subsequent sections we study the
effects of the torques on attitude motion.
Currently this section only includes the development of the gravity gradient torque.
6.1.1 Gravity Gradient Torque
We assume a rigid spacecraft in orbit about a spherical primary, and every differential
mass element of the body is subject to Newton’s Universal Gravitational Law:
Again, that rigidity issue…
In the integrand, the vector c~r is the only variable that depends on the differential
mass element. However, in general, this integral cannot be computed in closed form.
The usual approach is to assume that the radius of the orbit is much greater than
the size of the body, i.e., (formula didn’t copy well. Orbit >> body radius),
and expanding the integrand as follows:
BUT, what happens when the radius of the object is about the same as the radius of the orbit, as with the sun vs the barycenter motion? It would seem that you get to invent a new solution to the problem if you want to answer that question.
Again, my takeaway is just that there is something called “Gravity Gradient Torque”, it has an effect on orbiting things and changes their rotations, and it is very hard to ‘do the math’ other than in particular cases that have already been solved (small bodies orbiting big ones, and rigid).
Now, one other minor point: The gravity gradient comes out of the fact that some parts of both the earth and the satellite are nearer and further from each other. Yet for Jupiter vs the Sun, they are always a long ways away. I’m not able to say to what degree gravity gradient torque will act as though it were from the barycenter vs as though it were the J-Sun Distance. As Jupiter moves a very long ways in an orbit, it might well be that ‘one side of the sun vs the other’ is a large impact; or it might be that the sun spins so fast it all ‘averages out’. Need to “do the math” to know, and I’m not up to ‘doing that math’ as I have other things I care about more and it would take me a lot of time. Just realize, though, that you can’t just dismiss it with a “tides only” argument.
The rest of the paper gets a bit more hairy from there and folks with a masochistic streak are invited to read it…
Is somewhat more readable. It confirms the notion that this is a hard problem to solve, and will be worse for a magnetic ball of fluids with convection.
The interaction of three point masses under an inverse square law and
the rotation of a heavy asymmetric top about a fixed point are the two most
outstanding unsolved problems in Classical Mechanics. The solutions for
these problems are unknown in what is loosely called ‘closed form’, which
usually means the solution is specified or described by means of familiar
functions and/or afinite set of integrations of known expressions (quadratures).
Functional dependence of gravity gradient torque. -At a given point in a
gravitational field, a small body experiences a torque if the vector gradient
is not zero . For a spherical Earth field , the torque is given by the
which relates torque to ‘the unit vector from the Earth’s center’ and “The inertial dyadic of the satellite” whatever that is…
Completely unanswerable for me is how that same physics would apply to the sun wobbling about in space. I note in passing that the sun has a lot of inertia. The “I” term shows up as both a dot product and a cross product vs the radius and is not a divisor…
For amusement, you can listen in on a couple of folks trying to make a math simulation of the simple case of a small satellite orbiting earth:
I found this bit of ‘correction advice’ particularly interesting as it point toward an exchange of orbital angular momentum for rotational angular momentum, which is exactly what I’ve been trying to show happens on a macro scale (so SIM could ‘stir’ the sun).
If you do not take the orbital motion into account, then your spacecraft will just behave like a pendulum – the angular rate will vary periodically, which looks like what you are getting. The reason that the angular motion dampens out in orbit is because, as the orbiting body moves around the parent, the direction of the gravity gradient changes in the inertial frame. For your current model, the direction of the gravity gradient in the inertial frame is constant and, in a frame of reference rotating with the rigid body axes, the direction of the gravity gradient does not change any faster or slower than the angular rate of the orbiting body. This means that there is no mechanism for exchanging the angular momentum of the orbiting body with its orbital angular momentum, and therefore no damping mechanism.
The guy forgot to let the gravity come from a different direction as the thing rotated, so lost the mechanism to turn spin into orbital momentum. If I’ve read it right.
So it looks to me like he’s saying that as long as a satellite (the sun) is feeling gravity from changing directions (via the barycenter math construct or the real forces from the planets proper) it can convert between spin angular momentum vs orbital angular momentum. I think this matters. As it is a very real effect, causing all sorts of equipment to be put on satellites to control / use it; I think it would be very unwise to dismiss it with a wave of the Tidal Wand.
Now, what I’ve been singularly unable to find, is a simple explanation of just what mechanism causes it…
So this posting is more a ‘marking how far into the woods I got’ and not so much a map to the General Store. Perhaps someone else can move the map making a bit further than I was able to do.