# How Large Can a Planet be and Still Have Earth's Mass?

1. Dec 13, 2013

### Yae Miteo

Question: Just as in the title. How large can a planet be and still have the earth's mass? Obviously, this depends on its composition, just as long as it's solid and life can exist on it.

2. Dec 13, 2013

### Staff: Mentor

It depends on the density. Lower density means a bigger planet and vice versa.

3. Dec 13, 2013

### phinds

Your question is very vague. Define "solid". What "life" do you have in mind?

4. Dec 13, 2013

### Chronos

If a planet were composed entirely of ice [highly improbable], it would be less than 2 earth diameters in size.

5. Dec 13, 2013

I would call that a 'comet.'

6. Dec 13, 2013

### Yae Miteo

Okay, basically, "How large can a planet be and still be suitable for humans, i.e. not have an excessive gravitational pull. (no more than say, 10% greater than earth's)"

7. Dec 13, 2013

Newton's Law of Gravitation:
F = G*M1*M2/r^2
The surface acceleration of a planet (M2) is:
g = G*m2/r^2
but m2 = (4/3)*pi*r^3*d where d is the average density of the planet
so g = (4/3)*G*pi*r*d

Choose your favorite g and d and solve for r.

8. Dec 15, 2013

### snorkack

You cannot. d is NOT independent on g, nor r.

And no one knows the density of Earth!
Sure, the compressed one is known to three figures from Cavendish experiment. But the uncompressed one...

confidently claims 4,0

as confidently claims range 4,4 to 4,5

Thus, the compressibility of Earth is unknown by half - which should mean 25 % uncertainty in sound speed. Well, earthquake waves should be better measured.

You thus have no means of estimating the compressed density of a planet bigger than Earth. Even if Earth compressibility were known, you would have no idea what the compressibility does at pressures slightly higher than those present and observed inside Earth.

9. Dec 17, 2013

### Chronos

I fail to see how this is relevant, snorkack.

10. Dec 18, 2013

### snorkack

Highly relevant.
The question:
how big can an Earth mass solid planet be?
reduces to the question
how low density can Earth mass solid planet have?
And that depends on the compressibility of stone at high pressures.

11. Dec 19, 2013

### ViperSRT3g

I'm not as familiar as others are with the above math, but wouldn't you have much less compression of material in a decreased density planet?

12. Dec 19, 2013

Does a hypothetical planet HAVE to be stone?
Low density solids include H2O (917 kg/m3) - NH3 (817 kg/m3) - CH4 (423 kg/m3) - methane clathrate (900 kg/m3) - ammonium hydroxide (880 kg/m3) - all of which can be abundant enough in planetary space to form an earth-mass planet. These can be taken as an estimate to the lower limit of the density of a hypothetical solid planet.
These densities ignore compressibility (currently not known for most materials at the pressures developed inside an earth-sized planet). Compression will not change the mass of the planet, only its overall density, reducing the diameter.
Under extreme pressures, ice has a density of about 1300 kg/m3 (Ice XII, 800 MPa). A spherical body with this average density would have a radius 12.3% smaller than a spherical body of the same mass and the density of regular ice.
Rock is even less compressible than water or ice.