Orbital eccentricity variation of the other planets?

Orbital eccentricity variation of the other planets?

On Earth, it's fairly well published, mostly in climate change related articles, that the Earth's orbital eccentricity operates on a 413,000 year cycle with roughly 90,000-125,000 year variation within that cycle.


The cause of this variation is primarily the other planets, with Jupiter and Saturn usually mentioned as the primary causes (same link).

Question is two fold. How stable is the Earth's orbital eccentricity cycle? 413,000 years sounds enormously precise, but logically, I would think small changes in planetary orbits would create some variability. Is the 413,000 years well established and repeating or is it more uncertain?

And, do we have a good estimate of other planets orbital eccentricity variation? I looked, but couldn't anything at all on other planets eccentricity cycles. The closest I came was this gravity simulation chart below.

90,000 year eccentricity chart of the 4 inner planets here, and method used. Source.

You may take a look at the lates parametrization file by JPL-NAIF for the precession, nutation and pole orientation of the largest known bodies.

Although, for the large time scales you are asking, I expect you will need to propagate the data and make your own wild guess, or dig into appropiate literature about solar system physics.

Orbital eccentricity variation of the other planets? - Astronomy

We present the first secondary eclipse and phase curve observations for the highly eccentric hot Jupiter HAT-P-2b in the 3.6, 4.5, 5.8, and 8.0 μm bands of the Spitzer Space Telescope. The 3.6 and 4.5 μm data sets span an entire orbital period of HAT-P-2b (P = 5.6334729 d), making them the longest continuous phase curve observations obtained to date and the first full-orbit observations of a planet with an eccentricity exceeding 0.2. We present an improved non-parametric method for removing the intrapixel sensitivity variations in Spitzer data at 3.6 and 4.5 μm that robustly maps position-dependent flux variations. We find that the peak in planetary flux occurs at 4.39 ± 0.28, 5.84 ± 0.39, and 4.68 ± 0.37 hr after periapse passage with corresponding maxima in the planet/star flux ratio of 0.1138% ± 0.0089%, 0.1162% ± 0.0080%, and 0.1888% ± 0.0072% in the 3.6, 4.5, and 8.0 μm bands, respectively. Our measured secondary eclipse depths of 0.0996% ± 0.0072%, 0.1031% ± 0.0061%, 0.071%^<+0.029%>_<-0.013%>, and 0.1392% ± 0.0095% in the 3.6, 4.5, 5.8, and 8.0 μm bands, respectively, indicate that the planet cools significantly from its peak temperature before we measure the dayside flux during secondary eclipse. We compare our measured secondary eclipse depths to the predictions from a one-dimensional radiative transfer model, which suggests the possible presence of a transient day side inversion in HAT-P-2b's atmosphere near periapse. We also derive improved estimates for the system parameters, including its mass, radius, and orbital ephemeris. Our simultaneous fit to the transit, secondary eclipse, and radial velocity data allows us to determine the eccentricity (e = 0.50910 ± 0.00048) and argument of periapse (ω = 188.°09 ± 0.°39) of HAT-P-2b's orbit with a greater precision than has been achieved for any other eccentric extrasolar planet. We also find evidence for a long-term linear trend in the radial velocity data. This trend suggests the presence of another substellar companion in the HAT-P-2 system, which could have caused HAT-P-2b to migrate inward to its present-day orbit via the Kozai mechanism.

Spaceflight Mechanics

I.B.1 The Elliptical Orbit

The eccentricity of an elliptical orbit is defined by the ratio e = c/a, where c is the distance from the center of the ellipse to either focus. The range for eccentricity is 0 ≤ e < 1 for an ellipse the circle is a special case with e = 0. Semimajor axis a is positive for an elliptical orbit consequently, the total energy ξ is negative.

The extreme points on the major axis of the orbit are called the apse points. The point closest to the attracting body is called periapsis, while the farthest point is called apoapsis. For orbits about the earth, these extreme points are called “perigee” and “apogee,” respectively. True anomaly θ is measured from the periapsis direction such that θ = 180 ° corresponds to apoapsis [see Fig. 1 and Eq. (4) ].

FIGURE 1 . Geometry of the ellipse.

The period of an elliptical orbit (the time required for one revolution) is computed from Kepler's second law: the radius vector sweeps out equal areas in equal times. The constant “areal rate” swept out by the radius vector is dA/dt = h/2, where the constant h is the magnitude of the angular momentum vector. Separating variables and integrating over one orbital revolution yields the period P:

Equation (6) proves Kepler's third law, which states that the square of the period of an elliptical orbit is proportional to the cube of the semimajor axis a, which is the average of the periapsis and apoapsis distances.

The orbital eccentricity change of the Earth ?

We can so far I know not measure the distance to the Sun because the Sun does not reflect radar.

So how accurate is “observation” ?

Very accurate, actually. The trick is not to observe the Sun directly, but to observe the other planets, and to find a self-consistent orbital solution from these observations that agrees with General Relativity. One can even test for departures from GR and for unmodeled mass.

Once one does that, one can extrapolate the planets' motions several million years both forward and backward. When one does that, one finds that the planets' orbit orientations do loop-the-loops, and that their eccentricities and inclinations oscillate quasiperiodically. This leads to Milankovitch climate cycles on the Earth and likely also on Mars.

In particular, I've plotted

Eccentricity / Runge-Lenz vector:
Inclination / north-pole vector:

for the last several million years.

I've made videos for YouTube: [Broken] [Broken]

I got the numbers from here:
J. Laskar
"Secular evolution of the Solar System over 10 million years"
Astronomy and Astrophysics, 198, 341-362 (1988). 198..341L

I had to OCR the numbers and then painstakingly correct the OCRing, so there might still be some typos.

Resolving the Complex Evolution of a Supermassive Black Hole Triplet in a Cosmological Simulation

  • Matias Mannerkoski
  • , Peter H. Johansson
  • , Antti Rantala
  • , Thorsten Naab
  • & Shihong Liao

The Astrophysical Journal Letters (2021)

HD 143006: circumbinary planet or misaligned disc?

  • G Ballabio
  • , R Nealon
  • , R D Alexander
  • , N Cuello
  • , C Pinte
  • & D J Price

Monthly Notices of the Royal Astronomical Society (2021)

Higher-order effects in the dynamics of hierarchical triple systems: Quadrupole-squared terms

Physical Review D (2021)

Fast radio burst repeaters produced via Kozai-Lidov feeding of neutron stars in binary systems

Astronomy & Astrophysics (2021)

Orbital Dynamics with the Gravitational Perturbation due to a Disk

The Astrophysical Journal (2020)

Orbital eccentricity variation of the other planets? - Astronomy

Inner Planets: Mercury, Venus, Earth, and Mars,

Outer Planets: Jupiter, Saturn, Neptune, Uranus, and Pluto.

This simulation relies on a modern version of the mathematics of Kepler as developed and illustrated in Section 10.4 of the text.

Launch the Simulation

The sizes of the orbits and the orbital speeds are all to scale, but the sizes of the Sun and planets are not. As we know, Jupiter and Saturn dwarf all the other planets and the Sun is huge compared to everything else. To provide a sense of scale for the outer planets, the orbiting Earth appears in the center of the display.

The simulations correctly capture the fact that all the planets orbit the Sun in the same direction. But they model all the orbits on the orbital plane of the Earth and ignore the fact that the planes of the orbits differ from the plane of the Earth from about 1 to 3.5 degrees (except for Mercury’s orbit which deviates by 7 degrees). These differences are much greater for some comets and asteroids (some orbital planes are essentially perpendicular to the plane of the Earth).

Information for some of the more important comets and asteroids follows next. As before, a, , and T represent the semimajor axis in AUs, the astronomical eccentricity, and the period in Earth years, respectively. Kepler's third law tells us that a 3 /T 2 is the same constant for any body in orbit around our Sun. Since a = 1 and T = 1 for the Earth, it follows that T = a 3/2 for all planets, comets and asteroids. In some cases, the gravitational perturbations caused by Saturn and Jupiter result in irregularities, so that only averages or ranges for the orbital parameters can be given. The parameter i represents the angle between the plane of the orbit of the comet or asteroid and the plane of the Earth's orbit. As an illustration, consider Halley’s comet. The angle between the plane of Halley's orbit and that of the Earth is 17.8 o with Halley moving in the direction opposite to that of the Earth. Since 180 - 17.8 = 162.2, this information is captured by i = 162.2 o . In particular, i = 180 o would mean that the orbital plane is the same as that of the Earth, but that the motion around the Sun is in the opposite direction.

Comets contain matter left over from the formation of the solar system. Consisting of ice, dust, and gases, they are commonly referred to as "dirty snowballs." They are studied by astronomers (among other reasons) for the information they reveal about the formation process of the solar system. According to an explanation developed by J.H. Oort in 1950, comets originate in a cloud of comets, dust, and gases that lies in the range of 40,000 AU to 50,000 AU from the Sun.

Halley’s Comet: The English astronomer Edmund Halley (1656-1742) published his work on comets in 1705 in which he claimed that the comet of 1682 had a period of about 76 years and that it would be seen again in 1758. The fact that this proved to be correct was a glorious triumph for Newtonian mechanics. Today's information is more precise. Every orbit is slightly different and there is a variation of 17.62< a < 18.26. This corresponds to 74 < T < 78. The average values are a = 17.94 AUs and T = 76 years. The astronomical eccentricity is = 0.97. As already noted, the angle of Halley's orbital plane is i = 162.2 o . The most recent closest approach occurred in 1986.

Encke’s Comet: This comet was discovered by the German astronomer Johann Frank Encke (1791-1865). After having fought in the Napoleonic wars, Encke became director of the Berlin Observatory in 1825. The comet is named in his honor because he established its periodicity. The average values of its parameters are a = 2.22, = 0.85, T = 3.31, and i = 12 o . It has the shortest period of all known comets.

Shoemaker-Levy 9: This comet was first detected on March 24, 1993 by Carolyn and Eugene Shoemaker and David Levy in orbit around Jupiter. The comet had broken up into at least 18 large fragments after a close approach to Jupiter on July 7, 1992. The fragments impacted Jupiter in a spectacular way at the next closest approach during the period July 16-22, 1994. The available data allowed a reconstruction of the essential history of the comet. Prior to its capture by Jupiter, the comet orbited the Sun with a either in the range 3.5 < a < 4.5 AU (just inside Jupiter’s orbit) or in the range 6.0 < a < 8.0 AU (just outside Jupiter’s orbit). The eccentricity was in the range 0.05 < < 0.3 and the angle of inclination was in the range 0 o < i < 6 o . It is likely that the comet was captured by Jupiter during the years 1929 to 1939 and that it orbited unseen until its discovery.

Hale-Bopp: This comet was discovered in July 1995 by Alan Hale and Thomas Bopp. The average values of the orbital parameters are a = 187.48, = 0.99, T = 2567, and i = 89.4 o . The most recent closest approach to Earth was on March 22, 1997. It gained extraordinary notoriety at that time in connection with the mass suicide of the 39 members of a religious sect.

Hyakutake: Discovered by Yuji Hyakutake in January 1996 with a pair of binoculars, this comet had its most recent closest approach to Earth on March 25, 1996. The average values of its orbital parameters are a = 25.74, = 0.99, T = 130.6, and i = 124.9 o . This comet strains the simulation. Why do you think this is?

Asteroids are minor planets. Most of them orbit the Sun in the region between the orbits of Mars (semimajor axis 1.52 AUs) and Jupiter (semimajor axis 5.20 AUs) in what is known as the asteroid belt. All are relatively small, Ceres with a diameter greater than 940 kilometers is the largest, and only one, Vesta, is visible to the naked eye. It is thought that before asteroids could form into full-fledged planets, their orbits were tilted and elongated (possibly due to the gravitational effects of Jupiter) and that this prevented them from growing into planets.

Ceres: On January 1, 1801, the Italian astronomer Giuseppe Piazzi (1746-1826) pointed his telescope into the night in Palermo, Sicily, and noticed a very faint moving object. At that time, the Sun, the planets (with the exception of Neptune and Pluto), and some comets were the only known travelers in our solar system, so this was something new. Over a period of 41 days Piazzi observed the object move through an arc of 3 degrees across the sky. (This corresponds to about six Moon diameters.) But then it disappeared in the Sun’s glare. Was it possible to predict where it would reappear from the meager information that existed? To the 24 year old Carl Friedrich Gauss this question presented an exciting challenge, and he focused his extraordinary mathematical powers on it. Assuming that Piazzi’s object circumnavigated the Sun on an elliptical course, and using only three observed positions, Gauss estimated its orbit. The observations resumed, and on December 7, 1801, Piazzi’s object was relocated only a short distance away from where Gauss had predicted. This computational triumph brought Gauss immediate recognition as Europe's top mathematician and he became an instant celebrity. For more of the details, see

Eros: A small asteroid in orbit near the Earth. Studied by the space probe NEAR. ). Its average parameters are a = 1.46, = 0.22, T = 1.76, and i = 10.8 o .

Icarus: This asteroid has a highly eccentric orbit and travels as far as 1.97 AU from the Sun (beyond Mars) to as close as 0.19 AUs to the Sun (within the orbit of Mercury). Its average parameters are a = 1.08, = 0.83, T= 1.12, and i = 22.9 o .

Chiron: This asteroid/comet&rsquos orbit lies well outside the asteroid belt. It spends most of its orbit between the orbits of Saturn and Uranus. Its average orbital parameters are a = 13.63, = 0.38, T = 50.7, and i = 6.9 o .

Encke's Comet and the asteroids Ceres, Eros, and Icarus can be modeled using either the Inner Planets display or the Outer Planets display. The other comets and Chiron are best modeled with the Outer Planets display. Hale-Bopp will quickly go "off stage" in either display.

This simulation was created at the University of Notre Dame. Kevin Barry produced the original in 1997 and Michael Federico (ND '04) made the current update in 2003.

The Hydrogen-Like System

4.9.1 Slater Orbitals (STOs)

STOs were introduced long ago by Slater (1930) and Zener (1930) , and extensively used by Roothaan (1951b) in developing his fundamental work on molecular integrals. Slater showed that for many purposes only the term with the highest power of r in the hydrogen-like Rnl (r) is of importance for practical calculations. Zener suggested replacing the hydrogenic orbital exponent c = Z/n by an effective nuclear charge (Z – s) seen by the electron, and which is less than the true nuclear charge Z by a quantity s called the screening constant. This gives AOs which are more diffuse than the original hydrogen-like AOs. The Zener approach has today been replaced by the variational determination of the orbital exponents c, as we shall see in the next Chapter. One of the major difficulties of STOs is that the excited STOs within a given angular symmetry are no longer orthogonal to their lowest terms. As we shall see, this is particularly troublesome for ns orbitals ( n > 1), which lack the cusp which is characteristic of all s orbitals. Furthermore, multicentre integrals over STOs are difficult to evaluate.

Retaining only the highest (n – l – 1) power of r in the polynomial (58) defining Rnl (r), the dependence on l is lost and we obtain for the general STO in real form:

where N is a normalization factor. Here:

is the normalized radial part,

the normalized angular part in real form. As usual, we use Ω to denote the couple of angular variables θ, ϕ.

Separate normalization of radial and angular part gives:

so that the overall normalization factor for the general STO (116) will be:

with n = non-negative integer and a = real positive

The off-diagonal matrix element over STOs of the atomic 1-electron hydrogenic Hamiltonian is:

where Sn′l′m′,nlm is the non-orthogonality integral given by:

The diagonal element of the Hamiltonian is:

For c = z n (hydrogenic AOs):

where the last factor on the right is 1 only for the lowest AOs of each symmetry (l = 0, 1, 2,···), since in this case STOs and hydrogenic AOs coincide.

Orbital eccentricity variation of the other planets? - Astronomy

Why is Pluto so different from the other planets?

Aside from being, on average, the farthest planet from the Sun, Pluto and its orbit have several characteristics that make it unique. Its greater distance means its orbital period of 248 years is the longest of all planets. Its orbit has the highest eccentricity, which means that its distance from the Sun varies more than other planets. Its orbit is so far from circular that it can actually be closer to the Sun than Neptune at times.

The plane of Pluto's orbit is also tilted the most compared to the rest, taking it further north and south of the Earth's orbital plane than the other planets.

Pluto's only know satellite, Charon, is the largest satellite compared to the size of its mother planet. The Earth's moon held that title until Charon was discovered in 1978. Charon's large mass relative to Pluto means that the center of their common orbit about each other lies outside Pluto's surface, another unique characteristic of this Planet.

Finally, Pluto itself is unique for its position and physical characteristics. The four inner planets--Mercury, Venus, Earth and Mars--are known as terrestrial planets for their smaller size, solid surface, and similarity to Earth. The next four planets, looking outward from the Sun, are gas giants. They are larger than the terrestrial planets, have a larger number of satellites, and no solid surface. Beyond the gas giants, Pluto breaks the pattern by once again showing characteristics more like a terrestrial planet--small, solid surface, and only one (known) satellite.

Because of its peculiarities, many have suggested that Pluto should not really be considered a planet--that it had a different origin and is more closely related to a comet or asteroid.


The eccentricity of the Earth's orbit is currently about 0.0167 the Earth's orbit is nearly circular. Over hundreds of thousands of years, the eccentricity of the Earth's orbit varies from nearly 0.0034 to almost 0.058 as a result of gravitational attractions among the planets (see graph). [1]

Mercury has the greatest orbital eccentricity of any planet in the Solar System (e=0.2056). Before 2006, Pluto was considered to be the planet with the most eccentric orbit (e=0.248). The Moon's value is 0.0549. For the values for all planets and other celestial bodies in one table, see List of gravitationally rounded objects of the Solar System. Sedna the most distant known Trans-Neptunian object in the Solar System has an extremely high eccentricity of 0.85491 due to which it has an aphelion estimated at 937 AU and a perihelion at about 76 AU.

Most of the Solar System's asteroids have orbital eccentricities between 0 and 0.35 with an average value of 0.17. [2] Their comparatively high eccentricities are probably due to the influence of Jupiter and to past collisions.

Comets have very different values of eccentricity. Periodic comets have eccentricities mostly between 0.2 and 0.7, [3] but some of them have highly eccentric elliptical orbits with eccentricities just below 1, for example, Halley's Comet has a value of 0.967. Non-periodic comets follow near-parabolic orbits and thus have eccentricities even closer to 1. Examples include Comet Hale–Bopp with a value of 0.995 [4] and comet C/2006 P1 (McNaught) with a value of 1.000019. [5] As Hale–Bopp's value is less than 1, its orbit is elliptical and it will in fact return. [4] Comet McNaught has a hyperbolic orbit while within the influence of the planets, but is still bound to the Sun with an orbital period of about 10 5 years. [6] As of a 2010 Epoch, Comet C/1980 E1 has the largest eccentricity of any known hyperbolic comet with an eccentricity of 1.057, [7] and will leave the Solar System indefinitely.

Neptune's largest moon Triton has an eccentricity of 1.6 × 10 −5 , [8] the smallest eccentricity of any known body in the Solar System its orbit is as close to a perfect circle as can be currently measured.

Earth’s orbital dynamics

The tilted earth revolves around the Sun on an elliptical path. Over the course of a year the orientation of the axis remains fixed in space, producing changes in the distribution of solar radiation. These changes in the pattern of radiation reaching earth’s surface cause the succession of the seasons. The Earth’s orbital geometry, however, is not fixed over time. Indeed, long-term variations of the Earth’s orbit may help explain the waxing and waning of global climate in the last several million years. / Image by Thomas G. Andrews, NOAA Paleoclimatology

In the 1930s, Serbian mathematician Milutin Milankovitch theorized that slow changes in the way the Earth moves through space about the Sun could have influenced our planet’s climate past. The Earth has experienced a string of ice ages in the past, interrupted by shorter, warmer, interglacial periods. How –and how much– have the Earths’ orbital parameters (including eccentricity, obliquity, and precession) influenced global climate in the past?

Eccentricity describes the shape of Earth’s orbit around the Sun. Sometimes the path the Earth travels around the Sun is a nearly perfect circle other times, it travels an elliptical, or oval, path. A complex gravitational balance dictates that path the location and mass of other planets and celelestial bodies in the solar system can inflict gravitational forces on our planet.

Eccentricity measures how elliptical an orbit is. A nearly circular orbit has an eccentricity approaching zero (minimum e = 0.0005). An elongated elliptical orbit has a slightly higher eccentricity (maximum e = 0.0607). Earth’s orbit fluctuates between the two extreme values roughly every 100.000 years. Currently, our planet travels a fairly circular orbit (e = 0.017).

The path Earth travels around the Sun is important because it dictates how much solar radiation reaches the surface, and where. The Sun does not sit at the center of the elliptical orbit, so the Earth sometimes travels extra-close or extra-far from its energy source. When Earth’s orbital path is nearest to the sun, that point is known as perihelion. Currently, we reach perihelion in January when the large landmasses of the northern hemisphere are tilted away from the sun, making for more moderate winter temperatures. In contrast Earth reaches its furthest point from the Sun in July, making for less sweltering summer temperatures in the northern hemisphere. On the flip side of the coin is the southern hemisphere: near to the sun during southern summer, far from the sun during southern winter. Luckily, the expansive oceans of the southern hemisphere help moderate the more extreme temperature trends.

Seasons and the Earth’s tilt

We experience seasons because Earth sits at an angle. The tilted planet exposes more of one hemisphere to the Sun’s rays. As Earth orbits to the other side of the Sun, more of the other hemisphere is exposed to solar radiance while the first hemisphere experiences less sunlight. That’s why the seasons are reversed across the hemispheres… while the south bathes in summer light, the north experiences winter. If the Earth sat straight up and down and had no tilt, then there would be no seasons because light would strike the land at every latitude equally for the entire year.

Graphical examples of variations in orbital eccentricity (the shape of Earth’s orbit), axial obliquity (the degree that Earth’s axis tilts away from vertical), and polar precession (the slow conical ‘wobble’ of Earth’s axis). / Courtesy NOAA

Earth’s tilt is not always precisely the same. Every 41,000 years the tilt fluctuates between 22.1 degrees and 24.5 degrees from vertical. The tilt of the earth’s axis is described by obliquity. Earth’s current angle is about 23.45 degrees, that is, we’re tilted 23.45 degrees away from a hypothetical vertical line, and we’re very slowly becoming less oblique, shrinking the angle.

Those few degrees don’t seem like much, but if the planet is less tilted then the solar intensity that reaches the far reaches of the hemispheres is lessened. Insolation is a word for the amount of solar radiation received in a given area. When the axial tilt is low, the north pole and south pole receive less solar radiation overall, which lessens the relative strength of the seasons.

The precession of the equinoxes

The earth’s angle is fixed at 23.45 degrees. However, within a period of 22,000 to 26,000 years the tilted axis rotates very slowly. This precession has been described as a ‘wobble’ of the Earth on its rotational axis. Precession causes the timing of each solstice to advance slowly to a new point on the planet’s orbital path.

Make a fist with your left hand to represent the Sun, then hold a pen point-up with your right hand. The pen is the Earth’s axis. Over time, it moves in a circular motion, as though you were drawing a circle on the ceiling. Tilt the point toward the right, and orbit the pen around your fist without changing its orientation. That’s the Earth, now. Our axis points at the star Polaris, and when perihelion occurs (our closest approach to the Sun), the northern hemisphere is experiencing winter, which reduces the extremity of winter. Now, turn the pen’s tip through precession until it is tilted to the left. That’s the Earth, about 13,000 years from now. The axis will point at the star Vega, and when perihelion occurs, the northern hemisphere will be in the middle of summer… which means a hotter, more extreme summer.

What does it mean for climate today?

Next week, we’ll take a look at how well Milankovitch’s theories explain past ice ages, and how much these orbital processes impact modern-day climate change…

In the meantime, here’s a NASA graph of temperatures anomalies:

This map shows the difference in surface temperature in 2006 compared to the average from 1951 to 1980. Most of the globe is anomalously warm, with the greatest temperature increases in the Arctic Ocean, Antarctic Peninsula, and central Asia. NASA’s effort to track temperature changes will help societies evaluate the consequences of global climate change. / Map based on data from (NASA GISS Surface Temperature Analysis.) by NASA

Frontier Scientists: presenting scientific discovery in the Arctic and beyond

Watch the video: Ήχοι από τους πλανήτες (September 2021).