Planetary topology and gravity

[natski]

Dear all,

I am interested in the connection between the smoothness of a planet and the gravitational acceleration at the surface. Specifically, what is the highest a mountain can be for different values of g? More pertitently, what % of the Earth's surface would be covered with water if the value of g decreased or increased during it's intitial formation?

If anyone works on these questions, or knows something about it, or ideallly could list some relevant papers published onthe subject, I would be most grateful.

Regards,
Natski

 

[Chris Hillman]

One citation plus two warnings

Bad title: you mean topography, not topology.

I am interested in the connection between the smoothness of a planet and the gravitational acceleration at the surface. Specifically, what is the highest a mountain can be for different values of g?

Google for the superb textbook, available for free on-line from Cal Tech, Applications of Classical Physics, by Blandford and Thorne, and see Chapter 10, Exercise 10.3.

More pertitently, what % of the Earth's surface would be covered with water if the value of g decreased or increased during it's intitial formation?

What is this, "expanding Earth theory"? Look out! You should be aware that this is a crank "theory" [sic], beloved by the Creationism/Intelligent Design crowd, which runs counter to almost everything in geophysics, but which evidently suits some extra-scientific agendas.

 

[natski]

No I'm not familiar with the expanding Earth theory. I am trying to estimate the albedo of Earth-like planets. If the gravity on the Earth was greater, I hypothesise that the relief would be lower and hence the oceans would be spread thinner but further, significantly altering the surface albedo.

So what I'm looking for is some kind equation relating the variation in surface relief to the gravitational acceleration at the mean surface elevation.

 

[sneez]

Smoothness of a surface comes about different way than through gravity acceleration at that point (Errossions). Height of mountains to which they can rise has many factors, streess factors, crossectional base, slope of the slopes, density of surrounding base,internal composition, etc
There are calculation to which height a mountain can rise but gravity is just only one of the factors. I think your theory oversimplifies things which results in nonreality.

 

[natski]

Yes I agree that it is a simplification, but I don't believe it is a coincidence that Venus has an 11km peak with slightly lower g, and Mars has a 27km peak with significantly lower g. I also don't believe that Everest can grow (and sustain over geological timescales) much more than it's current height due to Earth's g, regardless of the tectonic activity below it. It may be a simplification but it could just act as a useful indicator to the upper limits.

 

[mgb_phys]

Mars has a 27km peak with significantly lower g. I also don't believe that Everest can grow (and sustain over geological timescales) much more than it's current height due to Earth's g,

Interesting point - I certainly learned that Olympus Mons was so high because of Mar's weak gravity.
But also that Everset is in isostatic equilibrium, it effectevifley floats on the crust rather than being held up by it. that implies the maximum height of a mountain should depend on the relative density of the mountain rocks and crust rocks - like an iceberg depends on ice/water density difference. Unless I'm missing something?

 

[sneez]

I am not geologists, but Mars and venus have no plate tectonics whereas Earth has. One of the factor that does also determine height of mountain is what kind of material is bellow the base as well as around the base into which it can slide or upon which it presses down.

Also, one cannot forget that less dense material can rise to higher heights. Since Mars is the least dense of them all its quite understandable that the peak is higher. IF you compare the mean density for Venus, Mars and Eearth and the peaks that you list very well support the theory that less dense -> higher height.

You should do some math which will straighten up how you think about it.

EDIT
:Ok here are the numbers: Venus 5240 kg/m3 , Mars 3900 kg/m3, Earth 5515 kg/m3

 

[D H]

Mount Everest is mostly granite. It literally floats on the much denser basaltic material that forms the oceanic crustal plates. The very early Earth was mostly basaltic and the variation in relief was much reduced. The early ocean covered the earth. The difference in density between non-hydrated rock (basalt) and hydrated rock (granite) is one of the critical factors that drives plate tectonics.

 

[natski]

So is anyone aware of any scientific papers on this subject?

 

[billiards]

Also, Venus and Mars have nowhere near as much water as Earth, this is crucial in rock mechanics, having less water means rocks are stronger which means you can build higher structures.

 

[natski]

To reply to Sneez... the density does indeed decrease as peaks increase but there is another effect at work here that is non-unrelated to my original hypothesis. The greater the value of g at the surface, the greater the compression of the planet's volume and hence the larger the value of the density. So it seems we have two re-encforcing effects to increase the height of mountains on low-g worlds- a) low g b) lower density

Does anyone have the equation for volume compression of planets on hand?

 

[sneez]

density=mass/vol

mass is proportional to Radius of planet ^3
g=GM/R^2 hence g is proportional to Radius^1

Note that Jupiter has 2.5g of earth, yet the density is only 1/4 that of earth. I urge you to put your theory on firm mathematical basis before speculating half-truths. There is lot of stuff on the net -> google is a very good start.

 

[natski]

density=mass/vol

mass is proportional to Radius of planet ^3
g=GM/R^2 hence g is proportional to Radius^1

mass is not proportional to radius^3 because the density, which you have assumed to be the constant of proprtionality, will not be constant, it will vary with radius since the material is under gravitaional compression. I believe only a calculus approach will adequately decsribe the problem.

A book by Hartmann seems to confirm this idea, 'Moons and Planets' (1999), where logarithmic curves of density against mass suggest a more complicated physical basis.

In any case, I am not looking to propose some radical non-matematical theory. I just want this idea as a tool for further work and am only playing with it rather than seriously proposing it. Furthermore, I would love to havre a matematical formulation of this but first the underlying concepts must be imagined. Ideally someone has already doen this work and I can merely quote their result.

 

[sneez]

Mass of a planet is proportional to radius^3 , because of spherical shape. Indeed, calculus is the only way to put something on firm mathematical basis. I just quickly assembled reasonable argument about g and density.

EDIT: I deleted the last end of the comment which was misleading.

 

[natski]

So does anyone know of an exact equation that relates the fractional compression of rock (i.e. the % increase in average density) to the mass, uncompressed mean density, and perhaps planetary radius?

 

[natski]

Perhaps one can apply the hydrostatic equation to the rock?

If dP=-g(r)*D(r)*dr

where P=external pressure, g(r)=gravitational acceleration at a radius r from the centre of the Earth and D(P)=density of surrounding rock as a function of external pressure

we know g(r)=GM(r)/r^2
sub in M(r)=4/3 * Pi * r^3 * D(P)

and we get:

dP=-(4/3)*Pi*G*r*[D(P)^2] dr

so we just need a function for D(P) which must increase as P increases and satisfy the boundary condition that D(P=0)=density of uncompressed rock. Any ideas for this function?

 

[sneez]

I think there are to be partials for density definitelly is function of temperature. Dont forget that for hydrostatics you are assuming ideal gas law :) Here my knowledge ends and no time to reasearch on this. I am not for hydrostatic equilibrium application to rock. The stresses and tensions and all that business does not make me easy using hydrostatics, but I have no time substantiate this. Well, good luck. PS: do goole search on "masss radius exoplanets" and you will 1st paper that could be relevant.

 

[D H]

natski, I think you are barking up the wrong tree here with regard to density. Rock is not particularly compressible (its a solid). That said, there will be dramatic changes in density as temperature and pressure transform the rock into different chemical compounds. In other words, density will be relatively flat as function of pressure except at phase boundaries. The biggest factor in planet density is composition, not pressure. I also think you are asking us to do far too much of your research for you, for free.

 

[natski]

I apologise D H, but no-one is obliged to answer my question, I am merely pursueing a line of thought in a scientific forum. In regard to compression fo rock, I discovered that the uncompressed density of rock on the Earth is 4400kgm^-3 whereas the average density of the Earth (compressed) is 5515kgm^-3 suggesting significant compression (http://www.newscientist.com/article/mg13017713.100.html).

 

[D H]

The average density of the Earth is much higher than that of continental rocks because the Earth's core is primarily iron. Short of stellar collapse, you cannot compress granite into iron with any amount of pressure.

 

[natski]

Just off google and a book I have got hold of, here are a few more references which support the compression of planets and the appropriate change in density...

http://www.solarviews.com/eng/mercury.htm

http://www.bookrags.com/Mercury_(planet)

http://www.psrd.hawaii.edu/Nov06/hit-and-run.html

Hartmann - 'Moons & Planets' 1999

It is interesting that none of these sources specifically say the compression is due to gravity or pressure, but I think we must logically assume a combination of the two factors is responsible and probably mostly pressure.

 

[natski]

Another really good reference is:

http://lasp.colorado.edu/~bagenal/3750/ClassNotes/Class15.16/Class15.html

specifically table 4.2:

http://lasp.colorado.edu/~bagenal/3750/ClassNotes/Class15.16/DensityTable.jpg

 

[mgb_phys]

It is interesting that none of these sources specifically say the compression is due to gravity or pressure, but I think we must logically assume a combination of the two factors is responsible and probably mostly pressure.

Gravity provides the force, pressure is just a measure of force per area.

 

[billiards]

I'm not sure what the point of this thread is, but I believe that there needs to be some kind of distinction between topography (which is a very "crustal" phenomenon) and the density of the whole earth, which is more-or-less unaffected by the crust. Also, pressure/temp equations are EOS (equation of state) thingamejigs, related to 18th century thermodynaims, with entropy and all that kuffaffall. A lot of research goes into these thing for different atomic structures to try to find minimum Gibbs energy for different structures using Schrodingers equations, I believe they're called ab inito methods. So yeah, pressure and temperature are vital to the state and hence the density of Earth materials.

 

[natski]

I found a brilliant paper on the matter of density by Valencia et al 2006 in Icarus. Have a look. :-)

 

[Chris Hillman]

natski, I don't know why you (apparently) blew off my suggestion, but I urge you again to follow up, since this is the first stop toward answering your question.

 

[billiards]

Also, I forgot to mention earlier, you should check out the Pratt and Airy models of isostacy.

 

[natski]

Thanks for the suggestions so far everyone, I didn't blow off your sggestion Chris, I've ordered the book! While I am waiting, I found further evidence to support my orginal idea... in the paper by Rivera et al 2005 on 'A 7.5 Earth-Mass Planet Orbitting the Nearby Star GJ 876', in the Discussion, page 20, bottom two lines, Rivera supposes...

'...the topography of the planet's surface, if it has one, is likely to be muted as a result of the high surface gravity (2.5 times that of the Earth) and the expected malleabilty resulting from the planet's large potential for retaining internal heat'.

If the planet's surface is being muted, than for any given volume of water on the surfac, we would expect a higher surface coverage and thinner depth.

 

[billiards]

Yes, it's a simple fluid mechanics problem, hardly anything new, a fluid (such as the Earth) will only be able to resist deformation from its lowest energy state (a sphere) as long as the fluid has the strength to resist the deformation. If there were enough water to cover the surface of the planet, the saturated rocks would be weakened significantly. This effect will make gravity's job a lot easier, or to put it less anthropomorphically, the rock will be less able to resist converting their gravitational energy into heat. So that if your planet did have significant gravity, then you would expect its surface to be relatively smooth.


The question as to whether the surface would be completely covered by water brings up other considerations. For the planet to have such a high g, it would need to be either incredibly dense, or incredibly large. Either way, it would probably collect a lot of space debris which could potentially evaporate away all the water by bolide impacts, these would certainly create fresh topography. Realistically, I don't imagine planets really get that much denser than the rocky planets (although I would be excited to be shown otherwise) - so the planet would have to be very large. Large planets tend to be gaseous, which would make the planet very hot which would evaporate the water.

Of course in real life we do have planets covered completely by water, or moons rather, completely covered by ice. Although I doubt very much that they have smooth topography.

 

[natski]

Nooooo!

Natski, you are obviously seriously confused about the meaning of "pressure", and it is now quite clear that one fundamental reason for that is that you have been blowing off the citation I gave you. The variation in both pressure and density is due to gravitation, period, full stop.
.

Actually I think I just didn't make my point very clearly. When I refer to gravity I refer to the variaton in the acceleration of gravity at a given distance r where as by pressure I mean the weight above any given element of mass within the Earth, which yes is given by gravity * mass.

It's really a moot point anyway since it's only a side remark. What I was trying to say si that as the weight of the mantle compresses the core (i.e. the internal pressure incerases), then the gravity at the surface will also increae since the radius has increased (hence the weight of the mantl has increased even more). So you have two effects both which add to the compression and which effect yields a greater % in compression is not immediately clear. This is what I was trying to convey.

 

[natski]

Realistically, I don't imagine planets really get that much denser than the rocky planets (although I would be excited to be shown otherwise) - so the planet would have to be very large. .

Actually the paper by Valencia, which has been heavily cited in recent Super-Earth papers, provdes an thorough and careful model that concludes significant compression of terrestial planets occurs.

 

[Chris Hillman]

We now return you to your regular broadcasting

It's really a moot point anyway since it's only a side remark. What I was trying to say si that as the weight of the mantle compresses the core (i.e. the internal pressure incerases), then the gravity at the surface will also increae since the radius has increased

Decreased. OK, sorry, I misunderstood.

 

[natski]

Yup decreased, sorry! :-)