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I saw the question ”Finding the mean distance between n points evenly distributed in a disc of radius r” and I thought that some of the constants may be of use to my problem.

My question is related to the pioneer 10 and 11 equipped with plaques which might be read by extraterrestrials. I wonder how close they would come to a star on its way out of the galaxy. Surprisingly there is little information about this on the net. You would think NASA had such things readily available since they agreed to create the plaque in the first place.

I assume that the probes continue in straight lines and disregard the galactic rotation as well as the aggregation of stars in the galactic center. I therefore turn to a simpler question to get some general idea:

What is the expected nearest distance to a star from a line through the galaxy perpendicular to the galactic plane. Suppose I project the stars into a circle with $10^{11}$ stars and take the mean value?

The quoted question used $7853981634$ $ly^2$

If I take that I get approximately $7853981634$ $ly^2/10^{11} $ to get the mean area around an average star, about $0.1$ $ly^2$ and therefore 0.3 ly between the stars as a mean distance from a probe to the closest star.

I thought about a distribution in 2 dimensions but I got discouraged by the complicated article Christian Blatter referred to in the other post. I realize there may be a duplicate somewhere but I can't see it right away.

Mikael Jensen
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1 Answers1

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Let's say we have a small disc $U$ of radius $r$ inside a big disc $V$ of radius $R$. If one drawn $N = \rho \pi R^2$ points $x_1, x_2, \ldots, x_N$ uniformly and independently from $V$, the probability that none of $x_i$ falls inside $U$ is

$$\verb/Prob/[ x_1, x_2, \ldots, x_N \not\in U ] = \verb/Prob/[x_1 \not\in U]^N = \left(1 - \frac{r^2}{R^2}\right)^{\rho \pi R^2}$$

If we fix $U$ and make $V$ larger and larger while keeping the density $\rho$ fixed. In the limit of $R \to \infty$, the probability that $U$ contains no $x_i$ becomes

$$\lim_{R\to\infty} \verb/Prob/[ x_i \not\in U, \forall i ] = \lim_{R\to\infty} \left(1 - \frac{r^2}{R^2}\right)^{\rho\pi R^2} = e^{-\rho\pi r^2}$$

Apply this to the problem at hand. If one pick a random point $p$ on the galactic plane and set $\rho$ to the local average of number of stars per unit area of galactic plane near that point.

The CDF of finding at least one star near $p$ within a distance $r$ will be $1 - e^{-\rho \pi r^2}$. So the expected distance of nearest star to $p$ is

$$\verb/E/[ r_{min} ] = \int_0^\infty r d[ 1 - e^{-\rho\pi r^2} ] = \int_0^\infty e^{-\rho\pi r^2} dr = \frac{1}{2\sqrt{\rho}}$$

Assume our galaxy has $N \approx 10^{11}$ stars and when we project them down to the galactic plane, they occupy a disk of radius $R \approx 50000{\,\tt ly}$ uniformly. We find

$$\rho \approx \frac{40}{\pi}{\,\tt ly}^{-2}\quad\implies\quad\verb/E/[ r_{min} ] \approx \frac14\sqrt{\frac{\pi}{10}}\approx 0.14 {\,\tt ly} \approx 8900 {\,\tt AU}$$

For such a distance with respect to the Sun, it is way outside the heliopause ($\approx 120{\,\tt AU}$), the theoretical boundary where Sun's solar wind dominates but still well inside the Oort cloud ($\approx 50000{\,\tt AU}$), the furthest structure that one can associate to solar system.

achille hui
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  • Thanks. Using the same galactic size and number of stars I came within a factor of (about) 2, although the quantity is perhaps not the same. – Mikael Jensen Dec 14 '16 at 08:10
  • The quantity isn't the same and a factor of 2 is way within astronomer's error margin. The weakest part of this estimation is probably the assumption that stars are uniform distributed across the disc. We know that there are more stars crowded near the galactic center. In any event, if one replace $\rho$ by appropriate effective density, any sensible estimation like what I or you do should get the same answer within an order of magnitude. – achille hui Dec 14 '16 at 08:30
  • Contact was lost with Pioneer at about 100 AU. After passing the next 100 galaxies (if the universe i big enough) the probe might come that close to a star and start to transmit again of the electronics works after a number of billion years, or alternatively the extraterrestrials have good abilities to detect anomalies within 100 AU. – Mikael Jensen Dec 14 '16 at 21:55
  • It now occurs to that the probes´velocity are much less than the galactic escape velocity. – Mikael Jensen Dec 30 '16 at 09:46