<p>Consider an initially stationary particle of matter and suppose a 1 Earth mass black hole flies past it at speed <span class="math-container">$v$</span> on a trajectory that passes the initial position of the particle at distance <span class="math-container">$r$</span>. The particle will be mainly affected by the gravity during a time period of roughly <span class="math-container">$r/v$</span> (up to some &quot;geometric&quot; constants), during which time it will accelerate at roughly <span class="math-container">$GM/r^2$</span>. This means that it will move a distance about <span class="math-container">$GM/r^2 \times (r/v)^2 = GM/v^2$</span>. So if this is small compared to <span class="math-container">$r$</span> it will be left &quot;more or less&quot; where it was, rather than being ripped away. So we can expect a hole of diameter a few times <span class="math-container">$GM/v^2$</span>.  <span class="math-container">$GM$</span> is about <span class="math-container">$4\times 10^{14}$</span> so to keep the hole diameter to say 1m you could need <span class="math-container">$v$</span> about <span class="math-container">$2\times 10^7 m/s$</span> which is about <span class="math-container">$0.07c$</span>.</p>
<p>We can also estimate the typical velocity change achieved by our test particle from this interaction, which is <span class="math-container">$GM/r^2 \times r/v = GM/rv$</span>, giving a KE per unit mass of about <span class="math-container">$$G^2M^2/r^2v^2$$</span>.</p>
<p>So, suppose the Earth has density <span class="math-container">$\rho$</span>  and radius <span class="math-container">$R$</span> the cylindrical shell of thickness <span class="math-container">$dr$</span> will mass <span class="math-container">$R\rho rdr$</span> (still ignoring &quot;small&quot; constants like <span class="math-container">$2\pi$</span>). That shell will acquire kinetic energy <span class="math-container">$G^2M^2R\rho dr/rv^2$</span> from the interaction. Integrating from <span class="math-container">$GM/v^2$</span> to <span class="math-container">$R$</span> we get a total energy deposited in the parts of the  Earth which are not &quot;ripped away&quot;. <span class="math-container">$$\frac{G^2M^2R\rho}{v^2}\log\frac{Rv^2}{GM}$$</span></p>
<p>Using <span class="math-container">$R = 6\times 10^6$</span>, <span class="math-container">$GM = 4\times 10^{14}$</span>, <span class="math-container">$\rho = 5000$</span>, and <span class="math-container">$v = 2\times 10^7$</span>  (all SI units) we get about <span class="math-container">$10^{25}J$</span>, not enough to actual destroy the Earth (Earth's gravitational binding energy is about <span class="math-container">$10^{32}J$</span>) but about <span class="math-container">$10^{10}$</span> megatons or an earthquake of about 13 on the Richter scale.</p>
<p>A black hole moving 1000 times slower (typical solar system velocity) would essentially destroy the Earth.</p>

CC BY-SA 4.0
2020-07-11T16:21:26.187Z
2020-07-12T11:44:21.937Z
2020-07-12T11:44:21.937Z
Steve Linton
<p>All this talk of <a href="https://arxiv.org/abs/1909.11090" rel="nofollow noreferrer">Planet 9 perhaps being a primordial black hole (PBH)</a> made me wonder how dangerous a PBH collision with earth would really be. </p>

<p>Specifically, if earth collided with a PBH of 1 earth mass, traveling fast enough that they didn't end up orbiting each other, what would happen?</p>

<p>Edit:
From a paper asking <a href="https://arxiv.org/abs/0710.3438v1" rel="nofollow noreferrer">Can one detect passage of small black hole through the Earth?</a> the analysis was that for an approximately proton-volume and mountain-mass sized PBH</p>

<blockquote>
  <p>"It creates a long tube of heavily radiative damaged material, which
  should stay recognizable for geological time."</p>
</blockquote>

<p>but is otherwise pretty hard to detect. </p>

<p>What would change if the PBH were the mass of the earth and the volume of a large marble? How slow would it have to be going before it really messed things up?</p>

CC BY-SA 4.0
2019-10-11T17:29:53.363Z
The tidal effects and the energy released by the accretion of part of earth's mass as it passed near and through the earth would be disastrous, although just how disastrous I can't say.  I'm guessing no atmosphere left and the remaining earth largely molten.
antlersoft
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2019-10-11T18:42:46.280Z
It would be apocalyptic
john doe
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2019-10-11T18:44:45.333Z
an earth mass black hole would have a diameter of about 1cm, rather less than a baseball (though qualitatively not much difference)
James K
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2019-10-11T18:46:47.523Z
john doe I don't think it would! If a mountain-mass collision is barely even detectable if you're standing right there, it seems to me it would be not so bad
joseph.hainline
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2019-10-11T19:28:49.370Z
@joseph.hainline at a distance of 640km, the force of gravity of the earth-mass black hole is 100 gees. At a distance of 200 km, the gravity from the earth-mass black hole is 1000 gees. And so on, by Newton's Law of Universal gravitation. The damage is going to be less than if an Earth-mass planet smacked into Earth, but it's still going to be ridiculously devastating.
notovny
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2019-10-11T22:12:52.710Z
You can find the parameters of a Schwarzschild  black hole using the [Hawking radiation calculator](https://www.vttoth.com/CMS/physics-notes/311-hawking-radiation-calculator). A 1 Earth mass BH is cold, about 0.02 kelvin. But gravity and tidal force in its vicinity are rather intense. And of course things are going to get rather toasty once it starts accreting stuff...
PM 2Ring
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2019-10-11T22:26:00.827Z
@notovny thanks, yeah I guess that pretty much answers the question. Don't think we'd survive that!
joseph.hainline
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2019-10-15T11:24:21.930Z
On the other hand, if the black hole is coming from the outer solar system, the encounter would be over in a fraction of a second. It is not obvious (to me) how much damage the gravitational (tidal) forces could do in such a short time scale. Similarly, this timescale seems short for the formation of accretion dynamics. I think my biggest worry would be the exchange of momentum between the black hole and the earth and the impact on the Earth's orbit.
TimRias
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2019-10-16T02:04:01.600Z
@mmeent I hope this isn't a dumb question, but can a BH "exchange" momentum? Can we even talk of "impact"? I'd have pictured it more as mass in the immediate vicinity of the BH being jerked into the bottomless gravity well as it slices at interstellar speed through the planet, and mass a bit less close getting jerked into the narrow cylindrical void the BH has just carved, and everything a bit more distant going "hey, what was that weird momentary heaviness thing?
Chappo Hasn't Forgotten
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2019-10-16T06:07:07.290Z
@Chappo: As far as you can talk about "impact" that does not do too much. However, the gravitational interaction of the black hole with the Earth will lead to some momentum transfer between the two bodies. I've not done the math in this case, but typically momentum is transferred from the heavier body to the lighter body. Given that the putative BH is ten times heavier than the Earth and travelling at pretty much the escape velocity for the solar system, even transferring 10% of its momentum to Earth could lead to the Earth being ejected from the solar system.
TimRias
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2019-10-16T12:03:48.453Z
@Chappo Coming in from the outer solar system would result in a relative velocity to earth on the order of 20-90 km/s. 90 km/s spends more than two minutes passing through the earth. Coming from interstellar speeds I probably wouldn't expect more than a few hundred km/s relative, as presumably the object is generally orbiting with the rest of the galaxy. Not a long period of time, but not a fraction of a second. That said, we're really getting into https://worldbuilding.stackexchange.com 's territory at this point.
notovny
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2019-10-21T07:23:56.777Z
@notovny Your gravity calcs in previous comment are spot on for BH of Earth mass (=1.77 cm diam.). Milky Way moves at 620 km/s relative to CMB rest frame, so using that as BH velocity (*waves hands*), and BH hits Earth's surface at 45º (=> travels 9000 km thru planet), the event takes <15 s. Is that too fast for much to fall in? At surface above midpoint, BH gravity is 1.2 gees; on opposite side of Earth, just 0.3 gees. We'd definitely *feel* its passage, but the maths to work out how much Earth mass is captured is way beyond me. Would [Worldbuilding.SE] cover that?
Chappo Hasn't Forgotten
CC BY-SA 4.0
2019-10-21T14:00:30.283Z
@ReinstateMonica are you confirming notovny's calculation that at 200km the BH would exert 1000 Gs? Because if so, that feels like more than enough to pull in solid rock and magma. That seems like it would punch a kilometers wide hole through the earth. 10 km wide? 100? Even 1 km hole through the earth is probably devastating. Would either of you be willing to put the math in as an answer to the question? I'm curious to see it.
joseph.hainline
CC BY-SA 4.0
2019-10-22T00:08:37.963Z
@joseph.hainline BH moving at 620km/s will only be within 200km of a particular point on its path for 0.7 s. *How far* can rock (crust/mantle) travel when being accelerated at 1000 Gs for <1 s? Key consideration is *limit on accretion rate*: see [this answer](https://physics.stackexchange.com/a/446221/113185) for relevant analysis. My non-physicist guess (the math is beyond me) is the 2cm BH leaves behind it a void in the order of maybe 500m diameter, which is immediately filled by the inrushing matter accelerated by the now-departed BH. Stupendous energy release - but just how disruptive?
Chappo Hasn't Forgotten
CC BY-SA 4.0
2019-10-22T01:31:24.503Z
Yeah but 1000 Gs is exerted at 200km at 100km it’s gotta be worse and so on. I’m thinking the hole is tens of km across.
joseph.hainline
<black-hole><planetary-science><collision>
How much damage would collision with a similar mass primordial black hole do to earth?
joseph.hainline
# How much damage would collision with a similar mass primordial black hole do to earth?

(Asked by: username_0 on 2019-10-10)

All this talk of [Planet 9 perhaps being a primordial black hole (PBH)](https://arxiv.org/abs/1909.11090) made me wonder how dangerous a PBH collision with earth would really be.

Specifically, if earth collided with a PBH of 1 earth mass, traveling fast enough that they didn't end up orbiting each other, what would happen?

Edit:
From a paper asking [Can one detect passage of small black hole through the Earth?](https://arxiv.org/abs/0710.3438v1) the analysis was that for an approximately proton-volume and mountain-mass sized PBH

> "It creates a long tube of heavily radiative damaged material, which
> should stay recognizable for geological time."

but is otherwise pretty hard to detect.

What would change if the PBH were the mass of the earth and the volume of a large marble? How slow would it have to be going before it really messed things up?

**Tags:** black-hole, planetary-science, collision
> **Comments:**
> **username_1**: The tidal effects and the energy released by the accretion of part of earth's mass as it passed near and through the earth would be disastrous, although just how disastrous I can't say.  I'm guessing no atmosphere left and the remaining earth largely molten.  
> **username_2**: It would be apocalyptic  
> **username_3**: an earth mass black hole would have a diameter of about 1cm, rather less than a baseball (though qualitatively not much difference)  
> **username_0**: john doe I don't think it would! If a mountain-mass collision is barely even detectable if you're standing right there, it seems to me it would be not so bad  
> **username_4**: @joseph.hainline at a distance of 640km, the force of gravity of the earth-mass black hole is 100 gees. At a distance of 200 km, the gravity from the earth-mass black hole is 1000 gees. And so on, by Newton's Law of Universal gravitation. The damage is going to be less than if an Earth-mass planet smacked into Earth, but it's still going to be ridiculously devastating.  
> **username_5**: You can find the parameters of a Schwarzschild  black hole using the [Hawking radiation calculator](https://www.vttoth.com/CMS/physics-notes/311-hawking-radiation-calculator). A 1 Earth mass BH is cold, about 0.02 kelvin. But gravity and tidal force in its vicinity are rather intense. And of course things are going to get rather toasty once it starts accreting stuff...  
> **username_0**: @notovny thanks, yeah I guess that pretty much answers the question. Don't think we'd survive that!  
> **username_6**: On the other hand, if the black hole is coming from the outer solar system, the encounter would be over in a fraction of a second. It is not obvious (to me) how much damage the gravitational (tidal) forces could do in such a short time scale. Similarly, this timescale seems short for the formation of accretion dynamics. I think my biggest worry would be the exchange of momentum between the black hole and the earth and the impact on the Earth's orbit.  
> **username_7**: @mmeent I hope this isn't a dumb question, but can a BH "exchange" momentum? Can we even talk of "impact"? I'd have pictured it more as mass in the immediate vicinity of the BH being jerked into the bottomless gravity well as it slices at interstellar speed through the planet, and mass a bit less close getting jerked into the narrow cylindrical void the BH has just carved, and everything a bit more distant going "hey, what was that weird momentary heaviness thing?  
> **username_6**: @Chappo: As far as you can talk about "impact" that does not do too much. However, the gravitational interaction of the black hole with the Earth will lead to some momentum transfer between the two bodies. I've not done the math in this case, but typically momentum is transferred from the heavier body to the lighter body. Given that the putative BH is ten times heavier than the Earth and travelling at pretty much the escape velocity for the solar system, even transferring 10% of its momentum to Earth could lead to the Earth being ejected from the solar system.  
> **username_4**: @Chappo Coming in from the outer solar system would result in a relative velocity to earth on the order of 20-90 km/s. 90 km/s spends more than two minutes passing through the earth. Coming from interstellar speeds I probably wouldn't expect more than a few hundred km/s relative, as presumably the object is generally orbiting with the rest of the galaxy. Not a long period of time, but not a fraction of a second. That said, we're really getting into https://worldbuilding.stackexchange.com 's territory at this point.  
> **username_7**: @notovny Your gravity calcs in previous comment are spot on for BH of Earth mass (=1.77 cm diam.). Milky Way moves at 620 km/s relative to CMB rest frame, so using that as BH velocity (*waves hands*), and BH hits Earth's surface at 45º (=> travels 9000 km thru planet), the event takes <15 s. Is that too fast for much to fall in? At surface above midpoint, BH gravity is 1.2 gees; on opposite side of Earth, just 0.3 gees. We'd definitely *feel* its passage, but the maths to work out how much Earth mass is captured is way beyond me. Would [Worldbuilding.SE] cover that?  
> **username_0**: @ReinstateMonica are you confirming notovny's calculation that at 200km the BH would exert 1000 Gs? Because if so, that feels like more than enough to pull in solid rock and magma. That seems like it would punch a kilometers wide hole through the earth. 10 km wide? 100? Even 1 km hole through the earth is probably devastating. Would either of you be willing to put the math in as an answer to the question? I'm curious to see it.  
> **username_7**: @joseph.hainline BH moving at 620km/s will only be within 200km of a particular point on its path for 0.7 s. *How far* can rock (crust/mantle) travel when being accelerated at 1000 Gs for <1 s? Key consideration is *limit on accretion rate*: see [this answer](https://physics.stackexchange.com/a/446221/113185) for relevant analysis. My non-physicist guess (the math is beyond me) is the 2cm BH leaves behind it a void in the order of maybe 500m diameter, which is immediately filled by the inrushing matter accelerated by the now-departed BH. Stupendous energy release - but just how disruptive?  
> **username_0**: Yeah but 1000 Gs is exerted at 200km at 100km it’s gotta be worse and so on. I’m thinking the hole is tens of km across.  

---

## 1 answers

### Answer by username_8 on 2020-07-11. Score: 2 | Accepted answer

Consider an initially stationary particle of matter and suppose a 1 Earth mass black hole flies past it at speed $v$ on a trajectory that passes the initial position of the particle at distance $r$. The particle will be mainly affected by the gravity during a time period of roughly $r/v$ (up to some "geometric" constants), during which time it will accelerate at roughly $GM/r^2$. This means that it will move a distance about $GM/r^2 \times (r/v)^2 = GM/v^2$. So if this is small compared to $r$ it will be left "more or less" where it was, rather than being ripped away. So we can expect a hole of diameter a few times $GM/v^2$. $GM$ is about $4\times 10^{14}$ so to keep the hole diameter to say 1m you could need $v$ about $2\times 10^7 m/s$ which is about $0.07c$.

We can also estimate the typical velocity change achieved by our test particle from this interaction, which is $GM/r^2 \times r/v = GM/rv$, giving a KE per unit mass of about $$G^2M^2/r^2v^2$$.

So, suppose the Earth has density $\rho$ and radius $R$ the cylindrical shell of thickness $dr$ will mass $R\rho rdr$ (still ignoring "small" constants like $2\pi$). That shell will acquire kinetic energy $G^2M^2R\rho dr/rv^2$ from the interaction. Integrating from $GM/v^2$ to $R$ we get a total energy deposited in the parts of the Earth which are not "ripped away". $$\frac{G^2M^2R\rho}{v^2}\log\frac{Rv^2}{GM}$$

Using $R = 6\times 10^6$, $GM = 4\times 10^{14}$, $\rho = 5000$, and $v = 2\times 10^7$ (all SI units) we get about $10^{25}J$, not enough to actual destroy the Earth (Earth's gravitational binding energy is about $10^{32}J$) but about $10^{10}$ megatons or an earthquake of about 13 on the Richter scale.

A black hole moving 1000 times slower (typical solar system velocity) would essentially destroy the Earth.


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