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I am wondering if anybody has ever calculated the odds of a rogue planet, which has been traveling through interstellar space and then enters into a galaxy, being able to travel all the way to the massive black hole at the center of that galaxy.

I would think that the odds of a rogue planet reaching the black hole at the center of the galaxy is pretty minuscule considering the gravitational pull of thousands or millions of stars and planets which it will likely encounter along its path as it is pulled towards the center of the galaxy.

Also, the closer it gets to the black hole, the more stars and planets it will encounter due to the distribution of stars in a galaxy with a black hole at its center.

What are the odds of a rogue planet that enters into a galaxy reaching the black hole at the center of the galaxy?

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    $\begingroup$ I think the best hope would be a simulation of many rogues, but remember this consequence of gravity being time-symmetric: if one object isn't even orbiting a second, gravity won't capture it unless one of both transfers energy or kinetic or angular momentum with a third. $\endgroup$
    – J.G.
    Commented Mar 30 at 15:18
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    $\begingroup$ I had similar questions in a different context. The links in my question might be of interest. How do rubble pile asteroids form? $\endgroup$
    – mmesser314
    Commented Mar 30 at 15:43
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    $\begingroup$ On a Fermi estimate, I'd wager something like probability of a rogue planet existing times the probability of the rogue planet aimed at our galaxy (possibly conditioned on the orientation of the original galaxy?) times the ratio of the BH radius to the galactic radius. The latter term alone is $\sim10^{-9}$ so it's almost certainly extremely small. $\endgroup$
    – Kyle Kanos
    Commented Mar 30 at 16:03
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    $\begingroup$ Note that galactic halos are growing, so if the planet came in with a small enough excess energy, it would not have enough to escape again due to the deepening of the potential well in the meantime. Then the question becomes about the likelihood of an already-orbiting object sinking into the center (and over what time scale?) $\endgroup$
    – Sten
    Commented Mar 30 at 16:54
  • $\begingroup$ What do you mean by "reach the black hole"? $\endgroup$
    – ProfRob
    Commented Mar 31 at 7:24

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The first thing to consider is that a galaxy is almost empty space and the visible solid matter is a tiny fraction.

Here are some 'back of the envelope' calculations. Taking the Milky way as an example, it is estimated to contain approximately $3\times 10^{12}$ stars though estimates vary quite widely. A typical star has around 1/3 the mass and radius of our Sun. (The radius of a star is roughly proportional to its mass for some odd reason.) The combined cross sectional area of the stars is then:

$$\pi \left(\frac{R_{Sun}}{3}\right)^2 \times 3\times 10^{12} \ \text{stars} = \pi \left(\frac{700000 \ \text{km}}{3}\right)^2 \times 3\times 10^{12} \approx 6\times10^{24} \ \text{km}^2$$

The radius of the Milky Way is estimated to about 53,000 ly so the cross sectional area of the galaxy is approximately

$$\pi(53000 \ \text{ly} \times 9.5\times 10^{12}\ \text{km}/\text{ly})^2 \approx {8\times 10^{35}} \ \text{km}^2$$

The ratio of cross sectional areas of solid matter to empty space is then approximately

$$\frac{6\times 10^{24}}{8\times 10^{35}} \approx 10^{-12} $$

or approximately one trillionth.

This is the ratio looking looking from above the disk plane but from the side the visible matter is not distributed as a sphere but more as a flattened disk like oblate spheroid. This increases the effective cross sectional area of the stars by a factor of maybe 20, looking from the side of the disk, but the proportion of the cross sectional area of the solid matter remains almost negligible.

This means that a planet entering like a projectile (with greater than escape velocity) is unlikely to collide with any stars and will probably pass through our galaxy on a curved trajectory, a bit like comets usually pass through the solar system without colliding with anything.

There is an assumption here that the planet was not initially on a course heading straight at the black hole at the centre. What are the chances of that?

The object (Sagittarius $A^*$) at the centre of our galaxy that includes a black hole has a Schwarzschild radius of approximately $1.2\times 10^7 \ \text{km}$ and wider radius of around $7\times 10^{12} \ \text {km}$ that includes the accretion disk and close orbiting stars. If we take the larger radius, the ratio of the black hole cross sectional area to that of the galaxy is approximately:

$$\frac{\pi (7\times 10^{12})^2}{8\times 10^{35}} \approx 2\times 10^{-10} $$

which again, is almost negligible.

Now lets consider the case of a planet that drifts in with low initial velocity. It will tend to be drawn in towards the black hole at the centre and acquire significant radial velocity as it falls inwards, but it will also acquire a certain amount of tangential velocity from the sling shot effect if it happens to pass near some orbiting stars (and possibly to lesser extent from frame dragging) and if it acquires sufficient angular velocity it may avoid falling into the black hole and end up orbiting it in a precessing elliptical orbit. This effect is difficult to quantify, but as mentioned above, the density of stars is so low, that the likelihood of passing sufficiently close enough to a star for the sling shot effect to be significant, is small and the probability of the slow moving planet being drawn into the black hole would be high in this case.

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  • $\begingroup$ If you squish the Milky Way down to a uniform disc of density 1 g/cm³, it would be quite thin. astronomy.stackexchange.com/a/41005/16685 $\endgroup$
    – PM 2Ring
    Commented Mar 31 at 3:08
  • $\begingroup$ Nitpick: "visible solid matter" – most of the matter we see in a galaxy is not solid, but plasma in stars, I think. $\endgroup$
    – Paŭlo Ebermann
    Commented Mar 31 at 10:15
  • $\begingroup$ @KDP, I like your answer, and it makes me wonder how many rogue planets from other galaxies have passed through the Milky Way galaxy over the past 13.7 billion years. $\endgroup$
    – user57467
    Commented Apr 4 at 0:53

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