# What would Happen if a Primordial Black Hole, with 5-10x time the mass of Earth, were to fall into our Sun?

So lately i heard of this theory that planet 9 might be a Primordial Black Hole (PBH) with 5 - 10 times the mass of Earth. I was thinking to myself, what would happen, if such a PBH (if it even exists) were to fall into our sun? Would it slowly consume the sun, even tough its mass is just a fraction of the star? Would the sun eventually become a black hole itself?

PS: Sorry for any typos or any wrong wording (I´m not a native english speaker)

PPS: I´m not a physicist or anything, i´m just interested in the universe and how it works.

A black hole accretes matter that gets too close, growing larger. However, (1) black holes have a very small radius, and (2) accretion tends to release a lot of energy that blows matter away from the vicinity.

The effective area of a black hole for gobbling up stuff is $$\sigma = (27/4)\pi R_s^2$$. For a 5 Earth mass black hole $$R_s=4.4349$$ cm. So the effective area is $$\sigma=0.0417$$ m$$^2$$. This is not much, about a letter paper. If matter from the sun's core flowed in at lightspeed it would gobble up $$1.8756$$ billion tons per second. This may sound impressive, but it would take 33 billion years for the black hole to eat the sun. One can refine this by taking into account that the hole would become larger, but there is also the problem that typical velocities will be way lower - likely by a factor of a thousand or more.

Second, as the matter swirls inward it releases a lot of energy in a bright accretion disk. This disk is so bright that the radiation pushes incoming matter away and the flow throttles. This is called the Eddington limit. For a 5 Earth black hole this is about $$2.62\cdot 10^{-4}$$ of the total solar luminosity, so it is not going to be very noticeable and the inflow gets slowed even more.

In short, dropping a planet-mass black hole will eat the star eventually, but it is such a slow process even for a biggish hole like this that the Sun will have time to become a red giant and a white dwarf before the hole actually begins to matter.

• That makes primordial black holes very difficult to detect. Seems that there could be a lot of them, and we would not know it. – S. McGrew Oct 7 '19 at 13:35
• @S.McGrew A ten earth mass object flying around in the inner solar system, would have a noticeable impact on dynamics of the Solar system. – mmeent Oct 7 '19 at 13:49
• Isn't that cross-section the cross-section for capturing particles travelling ultra-relativistically (i.e. the cross-section for photons)? This isn't appropriate for capturing particles in the Sun. The same answer you referred to suggests that the cross-section might be a million times larger. – Rob Jeffries Oct 7 '19 at 21:20
• This paper agrees with me. See eqn (3). The $27\pi r_s^2/4$ is the cross-section for photons or ultra-relativistic particles. – Rob Jeffries Oct 8 '19 at 22:36
• Also equations 1.1 and 1.2 in this paper that explicitly say the velocity is that with respect to the BH at infinity. royalsocietypublishing.org/doi/pdf/10.1098/rspa.2009.0331 The cross section goes as $c^2/v^2$. – Rob Jeffries Oct 8 '19 at 22:46

Anders Sandberg's answer assumed a black hole that somehow is already embedded in the Sun's core and stays there (despite travelling at an ultra-relativistic speed). In reality, a ten Earth mass black hole would fly straight through the Sun, hardly noticing that its there. Along the way it would "eat up" a tunnel of Solar material smaller than $$0.04 m^2$$, which would amount to about $$10^{-15}$$ Earth mass worth of material, which would only lead to a very minute perturbation of the black hole's orbit around the center of mass of the solar system.

If the black hole started on a bound orbit, this orbit will stay bound, and the black hole will pass through the Sun at each perihelium passage. However, it will take order $$10^{16}$$ such passages for the black hole to even double in size (let alone eat up the star). Given that observations exclude the presence of such a heavy object in the inner solar system, its period must be tens if not hundreds of years. So the time for it double in mass is going to be thousands to millions times the age of the Universe. This is also the timescale on which its orbit would start to change significantly due to backreaction of the absorption.*

Finally, we should remark that it is highly unlikely for the black hole to be on a trajectory that would even hit the Sun in the first place. It is much more realistic for the black hole to be on an elliptic (or hyperboloidal) orbit around the Sun, like all other objects in the solar system.

*Dynamical drag of the gas in the Sun likely has a much larger impact on the orbit, but is much harder to estimate. If the velocity is high enough it should still be fairly insignificant though.