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Scott Tremaine says here

..for practical purposes the positions of the planets are unpredictable further than about a hundred million years in the future because of their extreme sensitivity to initial conditions. As an example, shifting your pencil from one side of your desk to the other today could change the gravitational forces on Jupiter enough to shift its position from one side of the Sun to the other a billion years from now.

-- which is incredibly sensitive. Do our newly discovered gravitational waves alter the distribution of mass on Earth enough to have such an effect?

Qmechanic
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Yumquat
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    Not answering your very question, but just in case you are worrying if something might alter earth's orbit so that it will crash into the sun or get lost in outer space: may be, but gravitational waves are the least to be worried about. – Gyro Gearloose Feb 11 '16 at 21:45
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    And to your literal question "if the solar system is a sensitive chaotic system ..." it is unpredictable from start, so any additional marginal influence won't make a difference on account of predictability. – Gyro Gearloose Feb 11 '16 at 22:11
  • So could we say that there is an interaction, between planets and gravity waves, that influences their motion ? Don't worry, I'm not angling for astrology here. – Yumquat Feb 12 '16 at 01:14
  • The motion of planets is defined by the gravity field. Waves in the gravity field affects that motion, in proportion to the strength of these waves. It's good to realize that the waves are far, far smaller than the gravity of the sun and its planets. – MSalters Feb 12 '16 at 11:49

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The butterfly effect (which we are talking about here) is that small perturbations to the state of a dynamical system grow exponentially. More precisely and formally: If the states of two systems differ by $ε$ at a certain time $t_0$, at a later time $t_1$ they will differ by $ε·e^{λ (t_1-t_0)}$ (on average, and for sufficiently small $ε$). For most real systems, the exponent $λ$ (the largest Lyapunov exponent) is positive – tiny perturbations grow, and we have the butterfly effect.

As the solar system has a positive largest Lyapunov exponent, any perturbation will eventually grow so huge that it thwarts long-term prediction. This includes gravitational waves, as they obviously perturb the system (otherwise we couldn’t measure them). The only question is how long it will take for the perturbation to grow that large. If we know the Lyapunov exponent of the system, we can roughly estimate this. This is only roughly since the effect of a perturbation also depends on the nature of the perturbation – remember that the Lyapunov exponent is only quantifying the average.

Now the solar system is extremely complex, with each planet being its own chaotic subsystem. For example, we know that earth’s climate has a much higher Lyapunov exponents than the solar system and thus it is difficult to estimate the exact prediction horizon and impact of a particular perturbation. Therefore it is difficult to say whether the shifted pencil or the gravitational wave has a larger effect. If I had to wager a guess, I would say that the direct effects (such as the gravitational force between the pencil and Jupiter) are negligible, while the indirect effect via large-scale chaotic systems, such as the weather, dominates. As both perturbations grow to totally change these large-scale systems within an astronomically short time (say a year), I expect their effect on astronomical predictions to be comparable.

Wrzlprmft
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The example you give is <1 kg, <1 meter. Gravity waves move the whole earth (>10^24 kg) by a relative small distance (>10^-18). The product of that is well over a million times bigger.

MSalters
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  • So, the example Tremaine gives is a minute movement of the centre of mass of the Earth. Do gravity waves do that too, or would the warping of spacetime not mean a net shift in the centre of mass ? Would the whole planet move, or would it microscopically squish in places as the wave runs through it, like I've seen in some of the graphics about the LIGO discovery ? – Yumquat Feb 12 '16 at 01:17
  • @Yumquat: LIGO consists of 2 arms at a 90 degree angle. This is intentional: the gravity wave passing squishes matter in the direction of travel while expanding it in the other two dimensions (and vice versa, on the rebound). Gravity waves move at the speed of light, as far as we can tell, so they pass the earth in a few milliseconds, but the period of the decently detected wave was also a few milliseconds. In other words, the wavelength of those gravity waves was roughly the diameter of the earth. That's not "microscopically". – MSalters Feb 12 '16 at 11:45
  • The waves might be long, but I was asking about the effect on the earth, which I take it is a very slight deformation ? Or not ? – Yumquat Feb 22 '16 at 19:29
  • @Yumquat: It is very small indeed, about 1/1000th the size of an electron. – MSalters Feb 22 '16 at 21:12
  • Well that is.... pretty small ! But, what is the mass that is displaced ? Would it be as much as that of a pencil ? Or would it not actually displace mass but simply warp it in some strange relativistic way that i don't understand ? – Yumquat Feb 23 '16 at 21:34