# Frequency of nomad planets passing within 30 AU of the sun

A recent estimate by the Kavli Institute for Particle Astrophysics and Cosmology (a joint institute of Stanford and SLAC) is that there are circa 100000 times as many 'nomad planets' as stars

I found "The Close Approach of Stars in the Solar Neighborhood, Matthews, R. A. J., Quarterly Journal of the Royal Astronomical Society, Vol. 35, NO. 1, P. 1, 1994" which estimated that the frequency of other stars passing within a given distance to be

$$F_{r}(r) = \sqrt{2} \pi r^{2}\rho_{s}V_{s}$$

where

$$V_{s} \approx 19.5 \text{ km}/\text{second}$$

and $$\rho_{s} \approx 0.11 \text{ stars}/\text{parsec}^3$$

resulting in

$$F_{r}(r) \approx 10^{-5} r[\text{pc}]^{2} \text{year}^{-1}$$

Assuming that those estimates are accurate and substituting

$$\rho_{s} \approx 11000 \text{ planets}/\text{parsec}^{3}$$

and

$$r[\text{pc}] \approx 0.000145 \text{ parsecs}$$

we get a frequency of

$$F_{r} \approx (10^{-5})(0.000145^{2})(10^{5})/\text{year}$$

or

$$F_{r} \approx 2 \times 10^{-8}/\text{year}$$

This gives us a net 'close encounter' of the solar system with a nomad planet roughly every 50 million years.

Does this seem a reasonable estimate?

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Your LaTeX was doing exactly what you told it to do. There are a couple of ways to get upright text in mathmode. I applied one of them for you. You can see what I did by looking at the edit history. –  dmckee Feb 25 '12 at 19:25
No doubt these planets won't be spread about uniformly, but loosely clustered in streams, clouds or layers relating to surrounding star arrangements and galactic structure. They could be sparse in areas where stars swept by, and perhaps denser in the areas where stars are few, e.g between galactic arms. But with such a new hypothetical idea, one has to start with some sort of crude estimate to guide further thinking. –  DarenW Feb 26 '12 at 5:22

I decided to look at whether the estimate you arrive at gives a reasonable-seeming result.

0.000145 parsecs (30 AU, about the radius of Neptune's orbit) is a close encounter indeed. This closeness made me think at first that 50 million years seemed to often. We don't have evidence of giant planets passing that nearby that often.

So then I looked at the mass distribution of these 'nomad planets'. In the article it says that they are "ranging from the size of Pluto to larger than Jupiter", or $0.002 M_E$ to $>300 M_E$. We can assume that there are many more dwarf-planet sized bodies than gas giants.

Now we ask, does this seem right? About every 50 million years a Pluto-sized body passes as close as Neptune's orbit? And I have to say yes, this does seem reasonable. A body the size of Pluto would not produce much perturbation at that distance, almost certainly not enough to significanly effect the planets' orbits and only sometimes enough to disturb a few Oort Cloud and Kuiper Belt objects.

(A Jupiter-sized body might pass that close far less frequently - on time scales of a billion years or so; you could imagine that this could help explain some anomalies like the distribution of the giant planet's orbits, but that's just speculation.)

This is similar to how asteroids fairly commonly pass closer than the distance of the Moon to Earth. It's interesting and notable, but not catastrophic, which makes me want to say based on intuition that your estimate is reasonable.

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Further: since one can assume that there are many more Pluto-sized bodies than Jupiter-sized, then let's say then that the distribution is roughly logarithmic: we get on average a Pluto-sized body every 50 million years, a Moon-sized body every 150 million, a Mars-sized body every 350 million, an Earth-sized body every billion years. Doesn't sound unreasonable at all. –  Mark Beadles Feb 26 '12 at 15:57
The various micro-lensing studies in the '90s effectively measured the upper end of that distribution. No idea where to look or how easy the data will be to interpret. –  dmckee Feb 27 '12 at 17:47