(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?
Answer
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 $\left(m\right)$ and its orbital radius $\left(r\right)$:
$$U\left(r, m\right) = -mr^{-1},$$
in units where $GM_\textrm{Sun} = 1.$ The errors in $U$, one due to the error in $m$ and the other due to the error in $r$ will be
$$\delta U_{m} = r^{-1}\delta m \textrm{ }\textrm{ (magnitude), and}$$
$$\delta U_{r} = mr^{-2}\delta r.$$
The ratio of the errors is
$$\frac{r^{-1}\delta m}{mr^{-2}\delta r} = \frac{\delta m}{m} \frac{r}{\delta r} \approx 6 \times 10^{-14}.$$
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.
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