Etydhv

(This is going to be a strange question.)

How big a difference does the existence (or positioning) of a single human body make on the motion of planets in our solar system, millions of years in the future? I know we can't predict what the difference will be, but do we have reason to think that there likely will be a non-negligible difference?

Why might there be a non-negligible difference? Well, figuring out the motion of the planets in our solar system is an n-body problem. So that motion is supposed to be chaotic - highly sensitive to changes in initial conditions. At least, on a timescale of 5 million years, the positions of planets should be highly sensitive to conditions now. But just how sensitive is an n-body system to tiny perturbations in the gravitational field?

A single human body exerts some small amount gravitational force on nearby planets. So, if I add a single human to Earth's population now, or I move them from one position to another, how much would that change the motion of planets in the future? And over what timescale?

Bonus questions:

• Would the differences continue to grow over time, or would they eventually diminish to nothing? (I figure that in a sufficiently chaotic system they'd just keep growing, but would be interested to hear otherwise.)




• Would the effects be similar on the scale of a galaxy, or beyond?

Lasker published a well-known result in 1989 showing that the solar system is chaotic, the inner planets more so than the outer planets. Quoting the Scholarpedia article (written by Lasker himself):

An integration over 200 million years showed that the solar system, and more particularly the system of inner planets (Mercury, Venus, Earth, and Mars), is chaotic, with a Lyapunov time of 5 million years (Laskar, 1989). An error of 15 m in the Earth's initial position gives rise to an error of about 150 m after 10 Ma; but this same error grows to 150 million km after 100 Ma. It is thus possible to construct ephemerides over a 10 million year period, but it becomes essentially impossible to predict the motion of the planets with precision beyond 100 million years.

So one approach to your question, to get at least a qualitative answer, would be to compare a 15-m error in the Earth's initial position to a 70-kg error in its mass. Let's start with the Earth-Sun gravitational potential energy, which depends on the mass of the Earth $left(mright)$ and its orbital radius $left(rright)$:

$$Uleft(r, mright) = -mr^-1,$$

in units where $GM_textrmSun = 1.$ The errors in $U$, one due to the error in $m$ and the other due to the error in $r$ will be

$$delta U_m = r^-1delta m textrm textrm (magnitude), and$$

$$delta U_r = mr^-2delta r.$$

The ratio of the errors is

$$fracr^-1delta mmr^-2delta r = fracdelta mm fracrdelta r approx 6 times 10^-14.$$

You can use SI units to get the numerical result, but you don't have to plug in any values to see what is going on. Because the units cancel, we can just compare the ratio of the errors to the ratio of the values. The ratio of the errors, $delta m / delta r$, is approximately 5. But the ratio of the values, $r/m$, is $approx 10^-13.$

So, if the system's sensitivity to the mass error scales in a similar way to its sensitivity to the position error, it seems the mass error will have a much smaller effect than the position error for calculations covering 10 million years. Calculations that cover a longer period are not reliable regardless of the source of error.

Here is another thing to consider.

Let's model the sudden appearance of a random human on the face of the earth as a perturbation in the mass distribution of the earth. The magnitude of that disturbance will be of order

~(human mass/mass of earth)

which will be extravagantly small.

Now you scale the possible effect by the square of the distance between the source of the perturbation and the other planets of interest. This will be of order

~(extravagantly small number/enormous number squared)

Which we will call "teensy".

Next we compare the other sources of gravitational perturbation in the solar system, like the cycling of orbital positions between the various planets. These will be bigger than "teensy" by a factor of order ~(mass of human/mass of planet of your choice).

The upshot of this is simply that the other things which could affect the orbit of the earth are so much bigger than the gravitational effect of a single human that this will get completely lost in the noise.

@foolishmuse is absolutely right. If you are considering only the human population, there is no change whatsoever. The distribution is so random, that the shifted masses don't exert any change whatsoever. Of course you can cater to some fictional scenarios. Suppose our population grows a lot. We migrate a to other planets in fractions. Our multiple space exploration machines built over 5 million years will cluster. Here mass of the system is not conserved, since we will be using extraterrestrial resources too. There might be a change. Someone could do the calculations, based on the growth rates of population and our technological developments.