# Planetary topology and gravity

1. Sep 26, 2007

### 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 relevent papers published onthe subject, I would be most grateful.

Regards,
Natski

2. Sep 26, 2007

### Chris Hillman

One citation plus two warnings

Bad title: you mean topography, not topology.

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.

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.

Last edited: Sep 26, 2007
3. Sep 26, 2007

### 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.

4. Sep 26, 2007

### 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.

5. Sep 26, 2007

### 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.

6. Sep 26, 2007

### mgb_phys

Interesting point - I certainly learnt 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?

7. Sep 26, 2007

### 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

Last edited: Sep 26, 2007
8. Sep 26, 2007

### D H

Staff Emeritus
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.

9. Sep 27, 2007

### natski

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

10. Sep 27, 2007

### 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.

11. Sep 27, 2007

### 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?

12. Sep 27, 2007

### 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.

13. Sep 27, 2007

### natski

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.

14. Sep 27, 2007

### 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.

Last edited: Sep 27, 2007
15. Sep 27, 2007

### 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?

16. Sep 27, 2007

### 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?

17. Sep 27, 2007

### 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.

18. Sep 27, 2007

### D H

Staff Emeritus
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.

19. Sep 27, 2007

### 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).

20. Sep 27, 2007

### D H

Staff Emeritus
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.