The guest speaker at the February 2023 meeting of Keighley Astronomical society was Mr Rod Hine from nearby Bradford AS.
Mr Hine delivered a very pro active and demonstrative talk titled ‘Orbits, Spin and chaos’ which was very popular with the members who attended.
His was a very wide ranging presentation and here I have collated just a small aspect of the full lecture.
After the giant planet instability, the giant planets’ orbits kept spreading out ever so slightly due to planetesimal-driven migration. Meanwhile, the rocky planets finished their formation; during their “late accretion”, they continued to be bombarded for a billion years but at a rate that slowed in time.
For the past four or more billion years the planets have all been on roughly the same orbits as their present-day ones. It may seem kind of boring – but this doesn’t mean that nothing happens.
Imagine the planets following their orbits for billions of years. They zip around so fast that each planet blurs into a ring stretched out along its orbit. In time, these planetary “rings” jostle each other because of their mutual gravitational force. This doesn’t change the size of the rings — the average orbital distance of each planet remains unchanged. But this jostling does change the shapes of the planets’ orbits, that is, their orbital eccentricities.
Here is a visualization of five different orbits, all with the same average orbital distance but different eccentricities. The eccentricities of the Solar System planets are all between the circular orbit (in red, with eccentricity of zero) and the green orbit (eccentricity of 0.2, or 20%).
A planet on a circular orbit is always the same distance from the Sun. But on an eccentric orbit, the Sun-planet distance changes. The closest approach, or perihelion, is the average orbital distance times one minus the eccentricity. So for an eccentricity of 20%, the closest approach is 80% of the average distance. It’s analogous for the farthest approach, which is the average orbital distance times one plus the eccentricity. For an eccentricity of 20%, the farthest approach is 1.2 times (or 120% of) the average orbital distance.
Each of the planets feels the gravity pull of each of the others, and this determines the rhythm of its orbital evolution. This graph shows how the four giant planets’ eccentricities evolve over the course of two million years.
Each planet’s eccentricity bobs up and down over ten to a hundred thousand years. What this means is that the planets’ orbital shapes are changing in time. The amplitude of the eccentricity oscillation depends on how strongly each planet is gravitationally kicked by other planets. Jupiter’s eccentricity oscillates much less than Saturn’s, even though they are next-door neighbours (astronomically-speaking). This is simply because Jupiter is so much more massive than Saturn. And Uranus’ eccentricity bounces around a lot more than Neptune’s, because it is closer to Jupiter and Saturn.
Right now, Earth’s orbit is second-closest to a perfect circle of all the planets. Venus’ orbit is even more circular; the most-circular crown is traded between Earth and Venus. On a timescale of about a hundred thousand years, Earth’s orbit varies between an exact circle (eccentricity lower than 1%) and slightly stretched-out, with an eccentricity of about 6%.
You might think: 6% is pretty small. Those little variations can’t really matter for our climate, can they? (Spoiler – they do matter!)
In the 1920s, Serbian geophysicist and astronomer Mulin Milankovitch proposed that variations in the Earth’s orbital eccentricity and spin were indeed key drivers of Earth’s climate over long timescales.
The net effect of an eccentric orbit is to increase the temperature variations throughout the year. Compared with a planet on a circular orbit, a planet on an eccentric orbit passes both closer to the Sun at its closest approach (“perihelion”) and farther from the Sun at its farthest approach (“aphelion”). If Earth’s eccentricity was 20%, its closest approach would bring it 0.8 astronomical units from the Sun and its farthest approach would bring it to 1.2 astronomical units – a difference of 20% each way. The amount of energy received from the Sun depends on the inverse square of the distance to the Sun, so at closest approach it would receive 1/(0.8)2 = 1.56 times of the current value and at farthest approach 1/(1.2)2 = 0.69 times the current value – a difference of 30% to >50% compared with the average. But the relative difference between closest and farthest approach is big: the planet receives more than twice the energy each day it spends at closest approach compared with each day at farthest approach.
Earth’s spin changes in two ways. The axial tilt, or obliquity, oscillates between about 22 and 24.5 degrees every 41,000 years (the current value is 23.5 degrees). The direction that the spin axis points in space wobbles around every 26,000 years or so.
A planet’s obliquity affects which latitudes receive the most energy from the Sun. On a planet with zero obliquity, the spin axis points straight up out of the plane of the planet’s orbit. The poles receive virtually zero energy from the Sun, and the equator is maximally heated. As a planet’s obliquity increases, the poles receive more and more energy from the Sun. For a planet with an obliquity of 90 degrees, the spin axis is in the same plane as the planet’s orbit. The poles receive far more energy from the Sun than the equator. (This is the case for Uranus, which has an obliquity of 98 degrees).
The Milankovitch cycles are observed correlations between variations in Earth’s past climate and its historical orbit and spin. The image below shows how the different quantities are strongly linked, including ice ages. A particular quantity of interest is the amount of Solar energy received at a latitude of 65 degrees North, a particularly land-covered latitude. The different cycles associated with variations in Earth’s spin and orbital eccentricity appear to have dominated during different historical epochs.
Earth’s orbital evolution has therefore a pivotal impact on our climate, and one can easily imagine more extreme scenarios in which planets’ orbits have larger-scale oscillations that result in much larger climate variations.
On million-plus year timescales, it becomes apparent that the planets’ orbits are chaotic.
In a mathematical sense, chaos refers to a system whose future cannot be indefinitely predicted. The Lyapunov timescale indicates just how far into the future we can make a viable prediction. The shorter a system’s Lyapunov time, the “more” chaotic it is and the narrower the predictive window.
In a series of studies in the 1980s and 1990s, Jacques Lasker showed that the inner Solar System is chaotic, with a Lyapunov timescale of about five million years. The terrestrial planets’ orbits simply cannot be reliably predicted into the distant future. Small uncertainties in our knowledge of the planets’ exact positions prevents us from projecting their orbits into the distant future. These uncertainties are tiny – measured in meters or even millimetres.
Illustration of the chaotic nature of the rocky planets’ orbits. Their eccentricities are shown in a computer simulation that was extended for far longer than the age of the Solar System, backwards 10 billion years into the past as well as 15 billion years into the future. The curves show the maximum eccentricity in a given 10 million year-wide time window.
An additional complication in predicting the future of the inner Solar System comes from the asteroids. Several of the largest asteroids undergo close gravitational encounters, and our inability to predict the exact configuration and outcome of those encounters acts as a veil that blocks the possibility of predicting Earths orbital eccentricity more than 60 million years into the future.
In the image above the giant planets’ orbital eccentricities appear flat. That is, their orbital oscillations are consistent in time, so their evolution is regular and not chaotic. However, other researchers have come to the opposite conclusion, that the outer Solar System is indeed chaotic. Which one is correct?
The answer to this question was found in 2007. It turns out that we cannot know whether the outer Solar System is chaotic. This is because there exist both chaotic and regular solutions within the uncertainties in the known positions of the planets. Even though these uncertainties are tiny, only a few parts in ten million!
Even though Earth’s orbit is chaotic, its spin state is not. As we saw above, the tilt of Earth’s spin axis (its obliquity) only oscillates by about 2.5 degrees. It was shown in 1993 (also by Laskar) that this regular behaviour is thanks to the Moon. With no Moon, Earth’s obliquity would likely bounce around chaotically between zero and almost 90 degrees! But gravitational kicks from the Moon stabilize Earth’s spin axis and prevent this from happening, thus also stabilizing the Milankovitch cycles.
Chaos is a philosophical gateway. An orbit that evolves regularly can be predicted infinitely far into the future. A chaotic orbit cannot. This means that a system of planets evolving chaotically has the potential to become dynamically unstable, with planets on crossing orbits. A regularly-evolving system cannot.
He explained above that the inner Solar System is chaotic, and the outer Solar System may be. Does that mean they will go unstable in the future?
The planets’ orbital shapes, tilts and spins oscillate due to the gravity of the other planets — this has a strong effect on Earth’s climate. On billion-year timescales the terrestrial planets’ orbits are chaotic and cannot be predicted precisely.