Kozai-Lidov cycles

Given that I wouldn’t mind a break from the climate wars (although I don’t think I’m quite as cured as Stoat suggests he is), I thought I might write about something completely different. From 1958-1976, the Soviet Union had a fairly extensive Lunar programme. One of these missions, Luna-3, was the first to take a picture of the far side of the Moon (Note : not The Dark Side of the Moon). This particular satellite had a highly eccentric (elliptical) geocentric orbit and, after 11 orbits, is thought to have re-entered the Earth’s atmosphere, and burned up.

Kozai-LidovThat this was likely to happen had actually been predicted by a Soviet scientist called Michail Lidov. The reason was that the satellite’s orbital plane was highly inclined with respect to the Ecliptic (Earth-Sun) plane, and perturbations from the Sun and the Moon were causing the eccentricity and inclination of the satellite’s orbit to oscillate. If it hadn’t burned up in the Earth’s atmosphere, it would eventually have returned to its initial orbit. This is illustrated in the figure on the right, which is actually a test case from Naoz et al. (2011), but which I’ve reproduced myself. The plot shows 1 - e, where e is the eccentricity of the inner orbit, against time and shows that an outer perturber can cause e to vary from being very small (close to 0 – circular) to very close to 1 (highly elliptical). This effect was also described, at about the same time, by a Japanese astronomer called Yoshihide Kozai, and is often referred to as simply the Kozai mechanism. I think Kozai-Lidov is more appropriate.

WASP 4b - credit : Triaud et al. (2010)

WASP 4b – credit : Triaud et al. (2010)

So, why am I telling you about this? Well, there are two main ways to detect planets around other stars, more commonly know as exoplanets. One is to observe a star’s spectrum, to see if there is any shift in the star’s spectral lines. If there is, then you can use the Doppler effect to determine the radial velocity of the star. If the radial velocity shows some kind of cyclical (or sinusoidal) behaviour, then you can infer that something is in orbit about that star, and you can then infer the mass of this object, and some of the properties of its orbit. The other method is to simply observe a star and see if there are periodic dips in its brightness. If there are, you can infer that something must be passing in front of, or transiting, the star and, if it is periodic, it must be an object that is orbiting the star.

The top panel of the figure on the right, taken from Triaud et al. (2010), shows a typical radial velocity curve for a star that has a Jupiter-like planet orbiting with a period of just over 1 day (known as a hot Jupiter). In this system, the planet also transits/eclipses the star. If you zoom into the radial velocity curve near where the radial velocity is close to zero, you notice a kink in the curve. The reason for this is that the star is spinning as well as moving in an orbit. Typically, the spin of the star cancels (as much coming from one side as the other) but during a transit, the planet first blocks one side (making the star appear to be moving away from us faster than it actually is) and then the other (making the star appear to move towards us faster than it actually is). The shape of this kink tells us that the star must be spinning in the same sense as the planet is orbiting, just as it is in our own Solar system, and just as we’d expect if planets form out of discs formed from material with the same angular momentum as the material that formed the star.

WASP 17b - credit : Triaud et al. (2010)

WASP 17b – credit : Triaud et al. (2010)

Okay, so we’ve shown that an extrasolar planet orbits in the sense that we would expect based on conservation of angular momentum and based on our understanding of our own Solar system. Now look at the figure on the left. It’s the same kind of system as that we’ve discussed already (a hot Jupiter that transits its host star), but now the kink is opposite to what it is in the earlier figure. Now, when the planet transits, the star first appears to suddenly start moving towards us, rather than away, and then it appears to start moving away from us, rather than towards. What this tells us is that in this system, the planet is orbiting in almost the opposite sense to the spin of its parent star; very different to what we’d expect. The planet is in an almost retrograde orbit and, given conservation of angular momentum, we wouldn’t really expect planets to be able to form with such orbits.

So, how does such a planet form? Well, we think its the same basic process that caused Luna-3 to re-enter the Earth’s atmosphere and burn up after 11 orbits. We think that there must be a third body (probably stellar) on an orbit inclined with respect to the initial orbit of the inner planetary system which causes the inner system to undergo Kozai-Lidov cycles producing large variations its eccentricity and inclination. In this case, however, if it comes very close to its parent star, the parent star can tidally influence the planet, causing its orbit to shrink and circularize, but with a very different inclination to what it had initially.

So, there we have what is hopefully an interesting science story that goes all the way from understanding how the orbits of satellites can be perturbed by the Moon and the Sun, to explaining why some exoplanets appear to orbit their stars on planes inclined with respect to the plane on which we’d expect them to have formed.

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37 Responses to Kozai-Lidov cycles

  1. As an additional aside, the method for determining that inclination of a planet’s orbit relative to the spin of the host star was developed in the 1920s by two scientists, Rossiter and Maclaughlin and is known as the Rossiter-MacLaughlin effect.

  2. Eli Rabett says:

    For gigles, move about 50 light years out and predict what you would see looking at sol. With two/three gas giants and a couple of fair sized rocks it should be interesting

  3. Eli,
    At the moment we could detect Jupiter quite easily, as long as we observed for long enough (i.e., at least one Jupiter orbital period). Saturn – I think – we could also detect, but the orbital period is long enough that we wouldn’t have done so yet if we had started when this first became possible (~ 1995). The others would – I think – be undetectable with today’s technology unless they – by chance – happen to transit, but the possibility of this goes down as the orbital distance increases.

  4. BBD says:

    A bit OT, but are hot Jupiters thought to indicate a much older solar system than ours, in which gas giants originally formed at much greater distances from the star?

  5. No, systems with hot Jupiters can be almost any age and it’s thought that such planets initially formed at similar distances to where Jupiter formed in our own system. What I described here is only one way of forming them. They could form simply via inward migration in the disc, which means they’d reach their close-in orbit within 10 million years. They could be scattered by other planets in the system, which could also happen quickly. Or they can form as I describe here in which case the timescale depends on the period of the outer perturbed. Could take anything from a few million years to more than a billion years.

  6. BBD says:

    Thanks ATTP

    My question was particularly poorly phrased but you answered it anyway 😉 – gas giant formation is not thought to occur close to the star. Hot Jupiters moved inwards from their original orbits.

  7. John Hartz says:

    ATTP: How does your OP square with the Bible? 🙂

  8. BBD,
    Yes, exactly. There is essentially no known way in which they could have formed where they are now.

    I’m trying not to think of that 🙂

  9. ATTP: How does your OP square with the Bible?

    In which country?

  10. BBD says:


    My wife asked me why the orbits of gas giants would contract. I burbled about orbital velocity vs gravitational attraction between star and gas giant. She asked why the gas giant didn’t simply fall into the star and I said that I thought that its orbital velocity would increase as orbital radius decreased and a new quasi-stable orbit would eventually be established.

    How badly have I misled my wife?


  11. BBD,
    Roughly right. You need to conserve angular momentum, so in order for something to spiral in it needs to exchange angular momentum with something else (disc, other planet, additional star). It can spiral all the way into the star (in which case we would no longer detect it) but those that don’t can be left in a variety of different orbits, some of which are very close to the parent star.

  12. BBD says:


    Roughly right.

    Well, that’s a relief. I made a poor job of explaining that angular momentum of the GG must be exchanged with other stuff orbiting the star for its orbit to spiral inwards but this can be patched up. 🙂

    The fact that we can now detect extra-solar planets and infer somewhat about their mass and chemical composition is mind-boggling. As a kid, I used to dream about the day when this would be possible, without really believing that it would happen in my lifetime. Other worlds! And at last, there they are.

  13. WebHubTelescope says:

    Most interesting review paper is on moonSunEarth orbit by Gutzmiller

    Click to access Moon-Earth-Sin%20RMP.70.589.pdf

    He covers the impressive work by Hill in calculating the nonlinearities of the moon’s orbit.

    The work lead to the formulation of the Mathieu-Hill differential equation family. No doubt this has implications for earth sciences in general, as the simple concept of liquid sloshing is governed by a Mathieu equation.

  14. Everett F Sargent says:


    Are you referring to water waves?

    As I happen to know a lot of people who model water waves, including myself. 🙂

  15. WebHubTelescope says:

    First order perturbation to the shallow water wave equation. A similar perturbation that Hill discovered when trying to calculate ephemeris of moon’s orbit.

    The result in ocean of perturbation is of course much stronger as sloshing dynamics leads to even less predictable pseudo-oscillations. Think of ENSO.

    The math of sloshing is described on scales from liquid volumes on shipping containers to something much larger though the definitive research still remains.

  16. Andrew dodds says:

    BBD – look up the NICE model for the formation of the solar system.. Planets shifting orbits, swapping orbits, quite possibly being ejected entirely.. All sorts of fun.

  17. Andrew,
    It is an interesting model. I believe that one thing that the NICE model has been trying to explain is why Mars’s mass is so low. You would probably expect it to have been more massive. The explanation – apparently – is that Jupiter probably formed at 3AU and either used up, or ejected, a lot of planetesimals that would have formed Mars had Jupiter not been there. Jupiter then moved out to where it is now (5 AU). There are gaps in the asteroid belt (known as Kirkwood Gaps) that are indicative of Jupiter having moved, and forcing gaps to form at resonant locations in the asteroid belt.

  18. BBD says:

    Andrew – Thanks for the pointer.

    If anyone’s interested, here are two Nice model runs (animation and short discussion) suggesting two hypothetical evolutions of the solar system with orbits perturbed by exchange of angular momentum with Kuiper Belt Objects etc.

  19. Everett F Sargent says:


    Yes, well aware of ENSO and that type of ‘sloshing’ have been since the late 70’s (OTEC, stratified flows, Turner’s “Buoyancy Effects in Fluids” (the textbook I used back in the day)).

    As to this statement “The math of sloshing is described on scales from liquid volumes on shipping containers …”

    We once had 500,000 DWT liquid containers called supertankers, I had the 1st (and 2nd) PNA on my shelf for like two decades. I’m pretty sure naval architects have been dealing with ‘sloshing’ for at least a century now, the free surface correction to metacentric heights comes immediately to mind).

    I then got very interested in hydroelasticity (LMCS, RIBS) and left the field when they were heavily into RANS wave modelling (before that it was underkeel clearance in entrance channels and the Panama Canal (the 1997 ENSO), before that it was harbor resonance and moored ship motion).

    Haven’t picked up a copy of JFM in more than a decade now and I’m a relative peon to boot. .

  20. WebHubTelescope says:

    I gave it my best shot at either a Hill or Mathieu equation solution to ENSO here:

    Again the connection is that a third body in the earth-sun-moon system provides a perturbation to the periodic orbit. That is what Hill calculated.

    For the ocean, the induced sloshing creates a similar periodic perturbation that modifies the classical wave equation. I use the 4.25 year characteristic period that Clarke at FSU identified in the wave equation and then determined what the Mathieu/Hill perturbation sinusoids were that best fit the SOI profile of ENSO. Some rather obvious forcing functions such as the 2.33 year QBO play a significant role.

    Somebody had to do this model because it was a gap in the research as far as I could tell. Whether you buy into it, YMMV.

  21. Everett F Sargent says:


    Oh, and thanks for the link, I’ve added it to my collection of “Sun-Earth-Moon Tides” directory.


    What do you think of the works of Laskar?:


    Also along those lines, in relationship to GMSL and RSL, I’m considering the JPL orbital parameters as can be downloaded here:


    Or perhaps from this link:


    (so far I’ve only seen daily IERS data, I’d be looking for at least hourly data (e. g. JPL))

    Also, I see that a new leap second is coming up:


    Anyways, I’m very interested in the works of SONEL, satellite altimetry (AVISO, CU, etc), various ocean indicies (e. g. NINO), tide gage forensics (removal of ‘known’ tides (tropical year to minutes), relatively short duration hurricane/storm surges, etceteras) from the NOAA Tides & Currents website:

    http://tidesandcurrents.noaa.gov/inventory.html?id=8651370 (Duck, NC USACE FRF)

    I did some work on this ~2.5 years ago (MA/RI/CT//NY wrt Point Judith, RI) and am just now now looking at NC/VA (it does appear to be an interesting ‘puzzle piece’ as most of the sea level rise work (tide gages) uses only monthly data AFAIK).

  22. Everett F Sargent says:


    Have added your paper to my SEMT directory, very interesting, do you have plans for journal submission?

    Somewhat similar to some of the RSL work I’m currently interested in wrt NC/VA.


    I have a comment in moderation (has like 6 http links). TIA

  23. EFS,

    I have a comment in moderation (has like 6 http links). TIA

    Not that I can find.

  24. Rachel M says:

    I think I approved it already.

  25. Yes, of course. It’s the one before the comment where EFS pointed out there was something in moderation 🙂

  26. I’ve recently been wondering about the future evolution of our solar system.

    The Moon is spiralling away from Earth due to tidal friction, which slows Earth’s rotation. I used to think tidal braking would stop when Earth’s day matched the Moon’s orbital period, as with Pluto and Charon.

    Then I realized that even though this would stop lunar tidal friction, solar tidal friction wouldn’t stop until Earth’s day matches its year. Because lunar tides are about twice as strong as solar tides, lunar braking would probably finish first. That would lock Earth’s day to the Moon’s orbital period. Then that tidally locked Earth-Moon system would slow down and move away from the Sun until the Moon’s orbital period around Earth matches Earth’s orbital period around the Sun.

    I think this means the Moon would eventually end up in either the Sun-Earth L1 or L2 Lagrange points. Since Lagrange points are entries into the interplanetary transport network, that seems to suggest that the Moon would be more susceptible to orbital perturbations. Conceivably, the Moon could even be ejected from the solar system with much less energy than one would expect if it hadn’t settled into a Sun-Earth Lagrange point. (Right?)

    Even though our Sun would probably go red giant before this process finishes, a red dwarf with less than ~80% the mass of our Sun has an estimated lifetime longer than the ~13.8 billion year age of the universe. First generation stars seem to have been dominated by hypergiants and supergiants, but their bright flames died out quickly, seeding the cosmos with metals that may have helped smaller stars form. Some red dwarfs might be ~10 billion years old, and they’re still just infants.

    Furthermore, since tidal forces scale as the inverse cube of the separation distance, planets in closer orbits around red dwarfs would experience much faster tidal braking. An Earth-Moon system in the “habitable zone” of an ancient 0.3 solar mass red dwarf might have already established lunar and solar tidal locks long ago.

    Since most stars in the Milky Way are red dwarfs, and most recent surveys seem to suggest that most red dwarfs have planets, this could be a large source of moons which have been tidally locked into Lagrange points and potentially ejected into interstellar space.

    Is this potential source of rogue planets included in searches for baryonic dark matter? Even though CMBR anisotropies and abundant deuterium show that most dark matter is non-baryonic, projects like OGLE search for small fractions of baryonic dark matter like brown dwarfs using microlensing. Wikipedia claims that these searches have only excluded objects with half Earth’s mass and above, so they don’t seem to be sensitive enough to detect objects the size of our Moon drifting in interstellar space after being ejected from a red dwarf system.

    Unfortunately, I don’t know enough astrophysics to know if my musings make any sense, or alternatively if they were already considered decades ago.

  27. @DumbSci,
    I don’t think it is regarded as a mechanism for explaining dark matter. It’s true that planets around M-dwarfs in the habitable zone are more likely to be tidally locked. I don’t think anyone’s looked at what would happen to their moons, but they would be low-mass and so even if a substantial fraction were ejected, it’s hard to see how this could contribute much to the total mass budget. In fact, it’s likely that many planets are ejected through dynamical interactions with other planets or stars, but even this is probably not sufficient.

  28. Thanks Anders. Yeah, the total mass of all those ejected moons would be small, but only a tiny fraction of dark matter could even be baryonic in the first place. So I wasn’t sure how much mass would qualify as significant in that respect. You’re probably right, though.

  29. @DumbSci,
    I think the total mass of the Milky Way galaxy is about 10 times the estimated Baryonic mass. So, you’d need a lot of hidden Baryonic matter (brown dwarfs, ejected planets) if it were to make a significant contribution to the total mass, and I think that’s regarded as unlikely.

  30. Yes, but CMBR anisotropies and deuterium abundance show that most of that dark matter halo is non-baryonic anyway, so I wasn’t suggesting that ejected moons could make a significant contribution to the halo’s total mass. Instead, I was curious to see if this new (?) ejection mechanism could “significantly” increase ejection rates over the standard dynamical interactions with other planets and stars that you mentioned. The total mass of the Milky Way’s dark matter halo (which seems to be mainly non-baryonic WIMPs) seems less relevant than the much smaller estimated mass of MACHOs produced by standard dynamical interactions with other planets and stars.

  31. @DumbSci,
    Okay, I see what you mean. I don’t know that anyone has looked at this. There is quite a lot of interest in exomoons and it is actually possible to detect them, but this has focused on possible exomoons around more massive planets, rather than around the lower mass planets you might more commonly find around M-dwarfs.

  32. Rob Nicholls says:

    Really interesting post, and fascinating reading about Luna-3. Although I’d heard of the Russian Luna missions, I didn’t know any of the details. Discussion of orbits makes me wish I was better at maths. A few years ago I spent some happy hours writing a simple Newtonian 2D computer model of a planet with a ring (consisting of thousands of moonlets) and a single large moon. I was hoping that the moon (placed in the correct orbit) would make a gap in the ring through orbital resonance (like Cassini’s division in Saturn’s rings). The moon and the moonlets orbited the planet beautifully, but no gap appeared in the ring; not really surprising as I’d used very simple equations for movement and acceleration, so that each iteration or ‘move’ was a straight line move, (I’m sure there’s a way to calculate these things properly using integration but it’s beyond me); with each straight line move the moon / moonlets, moving at 90 degrees to the planet, would end up slightly further away from the planet than at the start of the move, with the result I had to fudge things and adjust velocity downwards at the end of each “move” so that total energy (kinetic plus gravitational potential) remained constant.

    Alas, no matter how hard I try, the hieroglyphs beyond fairly basic maths seem indecipherable, and it just seems awesome to me that there are people who can figure out how to calculate flight paths for probes that travel billions of miles and visit multiple planets, or land things on comets, or compress jpeg files etc.

  33. Rob,

    The moon and the moonlets orbited the planet beautifully, but no gap appeared in the ring; not really surprising as I’d used very simple equations for movement and acceleration, so that each iteration or ‘move’ was a straight line move

    Did you include the gravity of the Moon in your calculation? It sounds like you only included the gravity of the central planet, in which case the moonlets will orbit as if there is no moon present.

  34. Rob Nicholls says:

    I definitely included the gravity of the moon in the calculations, as if I made the moon’s mass big enough it would pull moonlets right out of the ring towards it, making interesting patterns (although for simplicity the thousands of moonlets did not exert any gravitational force). I think the equations were too simple, leaving a requirement for a fudge that possibly messed up the model. Hours of fun though. Maybe I’ll have another look at it.

  35. Rob,
    Interesting. Could be that the Moon mass just wasn’t high enough. Can also take a lot of orbits (> 100) before anything happens. If you know fortran, then the hermit code here is easy to use and very accurate (4th order).

  36. Rob Nicholls says:

    Excellent, thanks ATTP. I’ll check this link out.

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