# Why Can't Rocky Planets Exceed 14 Times Earth's Size?

Silverbackman

## Main Question or Discussion Point

A few weeks ago I saw a documentary that said that a rocky planets cannot exceed 14 times Earth's size, because any bigger than this it will collapse and cannot stand. Why is this?

Also, how many times bigger are Jupiter, Saturn, Uranus, and Neptune? I know they are much bigger than 14 times, but how come gas planets can this big and rocky planets cannot?

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I'm not sure why no one has replied to this thread, but I suspect it may be because we really don't have enough information on how planets form. Recent discoveries of gas planets orbiting close to parent stars suggest that theorys on planet formation may be inaccurate.

My guess is that, in very large planets, the pressure and tempature are too great to form solids. These planets may start out as rocky bodies, but as mass increases and tempature grow higher the rock is liquified or turned to gas which can't escape the gravity of the planet. Even rocky planets like the earth are largly liquid and gas.

I do not have an answer to offer for the alleged size limit of rocky planets. You have really perked up interest in finding an answer. There are so many unrezolved quetions about Solar System formation that it provides a compelling source of entertainment.

pervect
Staff Emeritus
My personal best guess was that above this limit, rocky planets would become brown dwarfs, in that the interior of the planet would be electron degenerate matter rather than a normal solid.

However, my confidence level was so low that I didn't want to post it in case someone had a better idea.

I don't have any hard figures or phase diagrams that describe the phase transition from normal matter to dwarf star (electron degenerate) matter.

russ_watters
Mentor
That would be my guess too, and the answer to the complimentary question of gas planets would then be that they have such a low density that the pressures inside do not make them become brown dwarfs until they are much, much larger.

SpaceTiger
Staff Emeritus
Gold Member
pervect said:
My personal best guess was that above this limit, rocky planets would become brown dwarfs, in that the interior of the planet would be electron degenerate matter rather than a normal solid.
A good guess, and I'm not an expert in planetary science, but I suspect that's not the explanation. My reasons for thinking this are:

1) The gas giants are thought to have rocky cores, in which there would be higher temperature and pressure than at the center of a ~20-30 earth mass rocky planet.
2) The planets are thought to have formed through accretion/collisions, so there would be nothing to stop such an object from forming. In other words, if you were right, theory should predict a bunch of tiny brown dwarfs scattered throughout the planetary systems in the galaxy. I've not heard of any such predictions.
3) This is mostly a semantic thing, but in the astronomical community, the definition of a brown dwarf has not yet been settled. However, the most popular ones involve either a mass cutoff (>~10 Jupiter masses) or a formation distinction (collapse rather than accretion). The hypothetical object in question wouldn't fit either definition.

My best guess is that the limit has something to do with atmospheric retention. That is, massive rocky planets could hold more gas than low-mass ones and there could be some threshold (perhaps ~15 earth masses) below which the outer gas is removed, either by solar wind or gradual leakage. This would mean that any rocky body above the mass limit could still exist, but only as the core to a more massive gas giant.

Any such theory would be speculative, however, and I don't have a lot of confidence in a limit like that. Planet formation is too complex to make any predictions with certainty, so I suggest we just wait and see.

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pervect
Staff Emeritus
I was thinking more along the lines that there might be a maximum radius to the planet, rather than a maximum mass. The original poster just said "maximum size", which isn't very specific, though I'd tend towards interpreting it as radius.

This would be based on the equations of state which control the size of a star which is not fusing. I think that this (size/radius of a non-fusing star) tended towards a constant? I'm not really sure I remember where I read that :-(.

SpaceTiger
Staff Emeritus
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pervect said:
I was thinking more along the lines that there might be a maximum radius to the planet, rather than a maximum mass. The original poster just said "maximum size", which isn't very specific, though I'd tend towards interpreting it as radius.
Ah yes, perhaps you're right. I sort of assumed it meant mass because that's the quantity astronomer's usually use describe planets. Also, I would expect the "maximum mass" to be in that ballpark as well. Well, let's check it. Rocky planets have a roughly constant density, so:

$$\frac{M}{M_{earth}}\simeq(\frac{R}{R_{earth}})^3\simeq 3000$$

3000 Earth masses is roughly 10 Jupiter masses, which is roughly the theoretical lower limit for brown dwarf masses. Seems to work, but I would still be curious to see the context of the OP's statement.

Phobos
Staff Emeritus
Gold Member
so let's contact the documentarians and ask them to back up that statement! :)

SpaceTiger said:
Ah yes, perhaps you're right. I sort of assumed it meant mass because that's the quantity astronomer's usually use describe planets. Also, I would expect the "maximum mass" to be in that ballpark as well. Well, let's check it. Rocky planets have a roughly constant density, so:

$$\frac{M}{M_{earth}}\simeq(\frac{R}{R_{earth}})^3\simeq 3000$$

3000 Earth masses is roughly 10 Jupiter masses, which is roughly the theoretical lower limit for brown dwarf masses. Seems to work, but I would still be curious to see the context of the OP's statement.
I saw this mentioned in an article concerning this issue, and it was indeed stated that 14 times the mass of the earth is what they referred to.

Also I don't think the cores of these gas planets are "rocky" any more than the earths mantle is "rocky" (maybe I am misunderstanding though as the lines between gas, liquid, and solid are somewhat counter intuitive). Accounts by theorists say that gas planets are assumed to be more "liquid" as you get deeper into the planet.

Not to try to answer for the OP, but I assume his info came from the same type of source.

It also seems that my explanation, and this from you;

My best guess is that the limit has something to do with atmospheric retention. That is, massive rocky planets could hold more gas than low-mass ones and there could be some threshold (perhaps ~15 earth masses) below which the outer gas is removed, either by solar wind or gradual leakage. This would mean that any rocky body above the mass limit could still exist, but only as the core to a more massive gas giant.
are saying essentially the same thing if I'm not mistaken.

SpaceTiger
Staff Emeritus
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GOD__AM said:
I saw this mentioned in an article concerning this issue, and it was indeed stated that 14 times the mass of the earth is what they referred to.
Okay, I would have expected as much. Planets are seldom described in terms of their radius.

Also I don't think the cores of these gas planets are "rocky" any more than the earths mantle is "rocky" (maybe I am misunderstanding though as the lines between gas, liquid, and solid are somewhat counter intuitive).
In my limited experiences with planetary science, "rocky" usually refers to chemical composition, not the physical state. Very poor terminology, I'll agree, but I think it's standard.

It also seems that my explanation, and this from you...
are saying essentially the same thing if I'm not mistaken.
I would say not, because I think the gaseous material is chemically and physically distinct from the "rocky" material that makes up earth-like planets. You stated that the gas would be formed in the very high temperatures and pressures of the rocky planet, indicating that even a rocky core could not exceed this mass. I was suggesting that rocky bodies could exceed this mass, but if they did, they'd be shrouded by a large gaseous envelope and be called "gas giants".

As I said, I'm not an expert in this field, so my guess could be wrong and yours could be right, but I think there is a difference.

Ok I understand. BTW the article went on to say that they weren't indicating these large "rocky" planets can't exist, just that it hasn't been observed.

SpaceTiger
Staff Emeritus
Gold Member
GOD__AM said:
Ok I understand. BTW the article went on to say that they weren't indicating these large "rocky" planets can't exist, just that it hasn't been observed.
That's strange. To my knowledge, we haven't observed any planets above one earth mass that were confirmed to be rocky...

SpaceTiger said:
That's strange. To my knowledge, we haven't observed any planets above one earth mass that were confirmed to be rocky...

I typed 14 times earths mass into google and got this

http://www.eso.org/outreach/press-rel/pr-2004/pr-22-04.html [Broken]

Havn't read it fully yet and of course can't vouch for the credibility, but it's interesting none the less.

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SpaceTiger
Staff Emeritus
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GOD__AM said:
I typed 14 times earths mass into google and got this

http://www.eso.org/outreach/press-rel/pr-2004/pr-22-04.html [Broken]

Havn't read it fully yet and of course can't vouch for the credibility, but it's interesting none the less.
That is interesting, but the problem is that the radial velocity technique doesn't give us the chemical composition (or even the physical size), so there's no way to know whether or not it's a rocky planet. However, they do say the following:

With a mass of only 14 times the mass of the Earth, the new planet lies at the threshold of the largest possible rocky planets, making it a possible super Earth-like object.
...indicating the existence of a theoretical limit in that range. Not that I would believe such a limit...

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SpaceTiger said:
...indicating the existence of a theoretical limit in that range. Not that I would believe such a limit...

Interesting that there is a "theoretical" limit with no "theory" (that I have been able to find in print) to explain why though. Not that my searches on google indicate the total knowledge of the scientific community. Still it is frustrating to say the least.

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I just did a quick search through some astronomical, physical and geological journals for things like "fourteen + earth + mass" and "rocky + earth mass + limit" etc... but nothing stood out. If anything has been written on the topic its either not in the popular journals, or wasn't taken seriously enough for many people to respond to.

pervect
Staff Emeritus
I did some more looking, and finally did run across a webpage which talked about the white dwarf star radius vs mass relation.

As I remembered, it was weird. Larger white dwarfs have a smaller radius! It's also not dependent on temperature.

http://www-astronomy.mps.ohio-state.edu/~ryden/ast162_4/notes17.html [Broken]

As the temperature T of the white dwarf's surface decreases, the radius R remains constant.

<snip>

There is an UPPER LIMIT to the permitted mass of a white dwarf. White dwarfs with larger masses have smaller radii.
So extrapolating a bit, if you pile more and more matter together, and it is of the sort of matter that can't fuse, you eventually expect it to form a dwarf of one color or other.

When it forms a dwarf, the radius will decrease as you add mass.

This implies that there is a maximum radius that you can reach, because after that point, when you add more mass, the radius goes down!

I don't have a good handle on what the numerical value of this maximum radius is at this point, nor do I have a firm handle on how much the chemical composition impacts the maximum radius. (I suspect the chemical composition isn't very important, but I'm not really sure).

If you keep adding matter, after you form a degenerate matter dwarf, you'll eventually form a neutron star, which will be even smaller.

http://imagine.gsfc.nasa.gov/docs/features/news/21sep04.html

for instance, talks about a neutron star that's 1.75 suns in mass, and has a best-estimate radius of 7 miles.

Keep on going (adding more mass), and you'll form a black hole. The size of the actual singularity will be zero, though one typically measures the black hole by the radius (surface area) of it's event horizon. (Measuring the size of the black hole by the event horizon size does mean that size increases as you add mass, but it would take a very large black hole to have a Schwarzschild radius of 15 earth).

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SpaceTiger
Staff Emeritus
Gold Member
pervect said:
I did some more looking, and finally did run across a webpage which talked about the white dwarf star radius vs mass relation.

As I remembered, it was weird. Larger white dwarfs have a smaller radius!

It's also not dependent on temperature.
Yes, in fact, the mass-radius relationship of white dwarfs is an excellent instructive tool because it can be very approximately derived by combining a basic knowledge of several areas of physics. In order to determine it, one must first consider the equation of state of degenerate matter. The pressure will depend only on the density of the material as long as:

$$kT << E_F$$

Why? Think crudely about the electrons as "trying" to fit themselves into a Maxwell-Boltzmann distribution, but failing because there are only so many states available in position-momentum space. Specifically, the exclusion principle limits them on the low-momentum end, so a degenerate gas will tend to fill up all of the states available between zero momentum and the fermi momentum. In practice, there will always be a high-energy tail, but one can approximately think of it as a filled sphere in momentum space; that is, the density is given by:

$$n_e=\int_0^{p_F}f(p)dp\propto p_F^3$$

Likewise, I can find the energy density of degenerate gas with simply:

$$U_e=\int_0^{p_F}E(p)f(p)dp$$

In general, the pressure and energy density are non-trivially related, but to a rough approximation, one can usually say

$$P_e \propto U_e$$

Given these things, we now have the tools necessary to derive a scaling relation for the equation of state; that is:

$$P_e \propto \int_0^{p_F}E(p)f(p)dp$$

There are two limits that are of interest: relativistic and non-relativistc. In the non-relativistic limit, one gets

$$E(p)=\frac{p^2}{2m}\propto p^2$$

In the relativistic limit, it is instead:

$$E(p)=pc\propto p$$

Substituting these into my above equation and combing with my first equation, we get:

$$P_e\propto p_F^5 \propto n_e^{5/3}$$ Non-relativistic degeneracy pressure
$$P_e \propto p_F^4 \propto n_e^{4/3}$$ Relativistic degeneracy pressure

What does all of that have to do with the mass-radius relationship? Well, imagine we combine this with some elementary gravitational physics. That is, let's recall hydrostatic equilibrium:

$$\frac{dP}{dR} \sim \frac{P}{R} = -\rho_e g = -\rho_e \frac{GM}{R^2}$$

$$P \propto \frac{n_eM}{R}$$

Plugging the equations of state into this and considering that

$$n_e \propto \frac{M}{R^3}$$,

we finally have the mass-radius relationships for non-relativistic and relativistic degeneracy pressure:

$$R \propto M^{-1/3}$$ Non-relativistic

$$M \propto constant$$ Relativistic

The first is the mass-radius relation you noted, and the radius does indeed decrease with mass. Notice that for the relativistic case, however, the mass/radius go to a constant. If derived in detail, it turns out that this will give you the famous Chandrasekhar mass!

DaveC426913
Gold Member
http://astron.berkeley.edu/~basri/defineplanet/Mercury.htm
Read the article to get the context, but these are interesting statements:

"...Nature produces objects with a continuum of masses, found in a variety of circumstances. However, we humans like to classify and name objects..."

"... an upper mass limit to planets (which therefore also constitutes the lower mass limit to brown dwarfs)..."

I would interpret this to mean that there is a continuum of object sizes/masses that range through planetary and up into stars, and that the deliniation between them is arbitrary.

SpaceTiger
Staff Emeritus
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"...Nature produces objects with a continuum of masses, found in a variety of circumstances. However, we humans like to classify and name objects..."

"... an upper mass limit to planets (which therefore also constitutes the lower mass limit to brown dwarfs)..."

I would interpret this to mean that there is a continuum of object sizes/masses that range through planetary and up into stars, and that the deliniation between them is arbitrary.
Yeah, that's what I was getting at in the last point of my first post. However, the 14 earth mass limit mentioned in the article GOD_AM is referring to is far too low to be the delineation for a brown dwarf. Also, it's not true that all astronomers view the brown dwarf/planet distinction as arbitrary. Some would say that the object should be named based on its formation mechanism -- collapse for brown dwarfs and build-up for planets. However, since there is no currently observable property that would tell us which is which, we usually go by mass.

What an interesting question. But the mass limit makes no sense at all. You will need a lot more than 14 times the mass of the earth to start a fusion reaction or a collapse into a neutron star.

I suppose you could get something like a bloated white dwarf? This would be what pervect means when he talks about electron degeneracy. It would be bloated because the size of a white dwarf is inversely related to its mass, so a white dwarf with a mass of only 14 times the earth must be pretty big. It seems like there should be a lower limit to the mass of a white dwarf but I could not find any, and 14 earth masses seems a bit low.

This possibility is very interesting to me because white dwarves take so long to cool down that even though these could cool down to yellow, red and black, I don't think the universe is old enough for any to have done so. However, if they could also be formed by large rocky masses then there could indeed be cool white dwarfs which are red and black since these would cool off quite a bit faster. In this case I would also wonder about their contribution to dark matter.

I find spacetiger's arguement for a radius limit to 14 earth radii using (M/Me)~(R/Re)^1/3 ~ 3000 to be pretty compelling. That is quite a coincidence. Maybe the source that GOD_AM refers to simply slipped up. To add weight to this arguement I would add that if it really was a mass limit then I cannot imagine them not mentioning what would happed if the mass was exceeded and what it would become. The berkely article relating an arbitrary upper mass to planets to the arbitary lower mass of brown dwarfs lends further weight to the likelihood that spacetiger argument is correct and that someone confused mass and radii.

SpaceTiger
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Gold Member
mitchellmckain said:
I suppose you could get something like a bloated white dwarf? This would be what pervect means when he talks about electron degeneracy. It would be bloated because the size of a white dwarf is inversely related to its mass, so a white dwarf with a mass of only 14 times the earth must be pretty big. It seems like there should be a lower limit to the mass of a white dwarf but I could not find any, and 14 earth masses seems a bit low.
More than a bit, I would think. The core of a dying star, even on the far end of the main sequence (~0.2 solar masses), would be much more massive than that.

This possibility is very interesting to me because white dwarves take so long to cool down that even though these could cool down to yellow, red and black, I don't think the universe is old enough for any to have done so. However, if they could also be formed by large rocky masses then there could indeed be cool white dwarfs which are red and black since these would cool off quite a bit faster. In this case I would also wonder about their contribution to dark matter.
As you say, I don't see any way for such an object to form. Besides that would be baryonic matter and to have it be the dark matter would be in contradiction with the CMB and nucleosynthesis.

I find spacetiger's arguement for a radius limit to 14 earth radii using (M/Me)~(R/Re)^1/3 ~ 3000 to be pretty compelling. That is quite a coincidence. Maybe the source that GOD_AM refers to simply slipped up. To add weight to this arguement I would add that if it really was a mass limit then I cannot imagine them not mentioning what would happed if the mass was exceeded and what it would become.
It's not entirely clear to me what would happen if such an object exceeded the brown dwarf size threshold either. The difference between planets and brown dwarfs is merely convention at this point. Besides, these folks are using the radial velocity method of planet detection, meaning there would have been no way for them to get a size for the object.

Wow this is old and I don't really know how a came across it but here's the answer. The gravity of a planet 14x's the mass of the earth overwhelms the electrostatic repulsion that comprise rock thus its impossible to have a rocky planet 14x's the mass of the earth.

I found this