# Celestial "orbits"

I heard that we're losing our moon, its slipping away from us in such tiny imperceptible steps that we don't notice any significant change even over kiloyears.

The same must apply to planets, inching away from the sun with time.

Now this means, from what I can infer, planetary orbits are actually spirals and not ellipses.

However, Newton's and Einstein's theories still work. My question is, I'm not quite sure as I would like to be, but circular orbits are incompatible with Einstein/Newton gravity theories. A circle is much closer to an ellipse than a spiral and yet the latter is explicable while the former is not.

Quomodo (how)?

• There's no answer to "deep" questions like this. Every theory is an approximation. Maybe in 100 yrs everything is some form of string theory (or some new totally theory), and we then know that using quantum methods, general relativity etc were just crappy approximations. As we speak, some physicists think MOND is correct is correct, to give but one example. "Newton's and Einstein's theories still work" that sentence is plain wrong. Newton was simply wrong, and it's like Einstein needs corrections (as you constantly hear, we have not resolved quantum V. relativity). And what about dark energy? Commented Nov 25, 2023 at 14:34
• "must" they? why? Commented Nov 25, 2023 at 16:53

Physical theories describe how things change in given circumstances (if the theory is right). In practice this means that they are applied in simplified ways, where the simplifications of circumstances are assumed to not be too severe so that the predictions will not have too large errors.

In Newtonian gravity two bodies orbit their center of mass in elliptic orbits. This is a valid simplification of how planets move around the sun that ignores the mutual gravitational influences of other planets: it becomes a bad approximation when the others get too close, have large masses, or if one let their influence last over very long timescales. Then a more complex model is needed for good predictions.

In the case of the Earth-Moon-Sun system the orbits are almost but not completely closed, yet it is rarely worth using the full three-body model in Newtonian mechanics (mainly since it doesn't have closed-form mathematical expressions): intermediate approximations are used to describe the actual motion to high precision.

Similarly in Einstein's general relativity two-body orbits are not ellipses and have a bit of precession (and, for bodies of comparable masses, we do not have closed-form expressions for them). This is again well modeled mathematically. One can also take gravitational wave emission into account, making them slowly spiral inwards. Here the mathematics of the underlying physical theory allows us to approximate how much this happens, and to numerically simulate the complex pattern of spacetime distortion.

So, do planets move in circles? To a first approximation, yes. But if you do careful astronomical observations, you will find them to move in ellipses (explained by Newtonian gravity). Even more detail, and you notice precession of these ellipses (due to oblateness, other planets, general relativity). Even more precision, and you will notice other gravitational deviation, as well as slow inspirals due to tidal forces and gravitational waves. These high precision details are small and hint at the complex interactions between parts of the world, even when underlying physical theories are known.

• It's just that the equations, from a drive-by, look so different. I can't parse the equation of a spiral. This kinda makes sense: $r = f(\theta)$ though, but that's in a polar coordinate system. What about in cartesian space? Commented Nov 24, 2023 at 5:55
• @AgentSmith - You can write spirals in any coordinate systems, but while an Archimedian spiral is easy in polar coordinates: $r=k\theta$ it gets more complex in cartesian coordinates $\sqrt{x^2+y^2}=k \arctan(y/x)$. en.wikipedia.org/wiki/Archimedean_spiral But these are the same curve. In physics we choose the representations that make the math easiest. But we also choose the underlying model of what curve or behavior there is depending on what fits the problem at hand best. Commented Nov 24, 2023 at 10:20
• $\sqrt {x^2 + y^2} = k \arctan \frac{y}{x}$ looks circlish-ellipsish. Completely forgot we can extract the angle from $y, x$ with the $\arctan$ function. Muchas gracias. So a circle is kinda sorta what Archimedes would've traced in the sand as he took his last breath ... as his hand went limp. 🙃 He could not, alas, complete the circle. Is $k$ the ever increasing radius? Commented Nov 24, 2023 at 11:33
• Also, based on a unit circle, shouldn't the $\arctan$ function cycle through the same values. How would the "radius" of the spiral increase, it would hit a wall (ceiling) i.e. its range is bounded. Commented Nov 24, 2023 at 12:20
• @AgentSmith The arctan function (like other inverse trig functions) is multivalued. arctan(a) = b + 2*pi*i, where i is any integer. Commented Nov 24, 2023 at 13:45

You are right. But you miss a few quirks of reality, or some fine-print of the applicable physics.

Yes, the Moon is moving away from Earth as slowing down Earth's spin rotation means that angular momentum has to go somewhere - and it only can go into orbital spin of the Moon which means it has to enlarge its orbit around Earth. Were its motion an exact circle, its motion would be an ever-so-slowly widening spiral.

In reality there is no such thing as a circular orbit - every orbit is elliptical to some degree, may it ever be so small. Thus the resulting motion is an ellipse which gets bigger ever so slightly

Further, general relativity implies a precession of the orbit around the center of gravity. It's a small effect, and it can be measured on Mercury's orbit, but it applies in principle to every other planet as well. That means that the position of perihel and aphel rotate slowly around the Sun, thus the orientation of ellips of the orbit around the Sun changes.

In practical terms, both, the widening of the orbit due to tidal action on the planets by the Sun as well as the orbital precession due to general relativity do not play a significant role, and one can neglect it usually. But if you really want to do high-precision predictions or calculations or predict orbits over billions of years, you need to take these effects into account - though predictions in systems with more than 2 bodies over extended periods of time have already their own problems with being chaotic (thus not exactly predictable due to uncertainties in the initial positions and small changes in the initial positions have big effect later on (butterfly effect)).

• Do planets other than Jupiter actually cause a tidal bulge on the sun, which would then cause such effects back onto the planets similar as our one does onto the sun? I'd suppose that the tidal bulge caused by Jupiter is by many orders of magnitude bigger than the one from earth, and its effect on earth would be bigger than the one caused by earth itself. Commented Nov 26, 2023 at 1:41

The model of the elliptical orbit is a close approximation of the reality—and even Kepler knew that, when he wrote that “orbits are ellipses.”

In reality, gravitational interactions between the Sun, planets, satellites, asteroids, comets, etc., make every orbit somewhat “spiraly,” but more along the lines of a not-carefully rolled-up string: there are parts closer to the main body, parts farther, etc.

For example, the Moon’s actual perigee (its closest point to the Earth) varies from about 356,400 to about 370,400 km. But if you were to trace its instantaneous orbit, you would get a different distance from whatever the last perigee was or what the next will be, because the Moon oscillates back and forth from the Earth even within the same orbit—by a very small percentage, but still.

If you read French, you may be interested in checking out http://astronomie.quebec/orbiteLune.php (or at least, look at its animations) that model—on the very small scale of a computer screen—the orbit of the Moon. https://ecliptiqc.ca/orbiteLune.php also gives an idea, though it’s not animated.