Largest possible size for a goldilocks zone planet with near Earth Gravity?

Lowkea

Hi All,

Very new to this forum, and hoping for some assistance.

I am in the planning stages for a science fiction sequence of books. I am not planning on writing true 'hard' sci-fi as unfortunately I do not have the knowledge base required, however I would like the world building in my novels to be theoretically possible.
I am very taken with the idea of the 'super earth' I have read about the relatively recent discovery of Kepler-22b and it really sparked something inside me.
The greatest problem I see straight at the outset is that Kepler is estimated to have a gravity 7 times that of Earth, obviously not the most hospitable of environments for Homo Sapien! To my understanding gravity is influenced by mass, density and speed. (please forgive my ignorance I am more a humanities kinda guy!)
My question is what is the largest possible size planet you could have that has, say, a maximum of 1.3 times Earths' gravity, while still having enough density not to collapse into itself?
I would be very interested to hear any other possible problems that could be foreseen with a 'super earth' type planet capable of supporting human life.

BadBrain

It depends upon what you mean by "near-earth gravity".

The gravity equation is F = G(m1m2)/r^2, where F is the force of gravity, G is the gravitational constant, M1 is the mass of one attractive/attracting body, m2 is the mass of the other attractive/attracting body, and r is the radius of orbit of the orbiting body around the orbited body. The overall force of gravity on a planetary surface doesn't need the m2 part, so it's the mass of the planet times the gravitational constant, the whole divided by the radius from the planet's center to its surface. In determining the relative gravitational pull at the surface of two different planets, you divide the equation for the one planet by the equation for the other, in which process the gravitational constant and, for a loose approximation, the radii, cancel out, leaving the entire equation as the proportion of the masses of the two planets. Thus, the moon has 1/6 the mass of Earth, and the force of gravity on its surface is 1/6g. Mars has a little less than 40% of Earth's mass, so gravity on Mars' surface is about 40%g. If you want to have the planet's density equal to Earth's, then adjust the radius from the center of the planet to its surface. If you want the planet's density to be different, but the volume to be the same as Earth's, then just adjust the mass. (Remember that the moon has 1/4 Earth's volume but only 1/6 Earth's mass).

Hope I helped.

Ryan_m_b

Welcome to the forums this isn't my field but I think I can give you some pointers.
Gravity is determined by mass but density should be taken into consideration when thinking about surface gravity. For example: the orbit of a probe orbiting Earth at 30,000km will have no affect if the Earth halved or doubled in size but maintained a constant mass. However the surface gravity of the smaller Earth would be greater whereas the larger Earth would have a lower surface gravity.
Not sure what you mean by the last bit, the more massive an object is the more dense it becomes as gravity acts to pull everything in. I don't know how to answer your question directly as I am unsure of how to work out the surface gravity of an object given its size and density (if another member reads this and has a helpful equation please post!) however an easy way to postulate a bigger planet with the same surface gravity would be to make it larger and less dense. To do that simply make it out of a different material, for example Earth is iron but a planet made of silica totally would be less dense (though I'm unaware of the likelyhood of this).
Well without a terrestrial biosphere it wont be supporting any humans although you could postulate them living in artificial closed ecological systems. If you go with the latter option then I can't think of any problems but if you are supposing that this planet would have it's own/a terrestrial biosphere then you'll have to contend with slower days (unless it has a faster spin which means stronger coriolis effect which means stronger winds), increased energy entering the system from the larger surface area (possibly leading to extreme weather), greater disparity in climate and therefore ecosystem between lattitudes and (if you go with the lower density materials) lack of heavy materials like metals.

Hope that helps!

EDIT: Ah it seems badbrain beat me to it! Just to use his equation for Earth that would be roughly (6e24 x 6.67e-11)/6.35e6². So working it through if you double the radius of Earth you increase the volume by 8x and would need to increase the mass by 4x to maintain the same surface density.

Lowkea

I guess that is my problem, and i apologise as I know my ignorance is shining forth almost blindingly right now :P
Of what conceivable/practical material could this planet be made of to ensure that it does not collapse into a more dense object. Any suggestions?

Your points about the biosphere are really valid too. I am envisaging an environment that can support its own life and also the introduction of humans. Any ways you can think that I could possibly mitigate the increased coriolis effect to create a viable ecosystem that is within the realms of possible science? Although a longer day may be the way forward.

Ryan_m_b

I'm not sure how to work that out, I guess one would have to take into account the compressive strength of a material. As for a suggestion IIRC aluminium is half the density of iron and carbon is a quarter so that might be a starting point.
Hmmm I'm highly skeptical of the idea of an alien ecosystem that humans can integrate into. Firstly because even if they didn't use alternative biochemistry and the environmental factors (atmosphere pressure/content, soil/water content etc) were the same it's highly unlikely they would have evolved in a way that allows us to eat them and survive/get any nutrition. And if this ecosystem had have evolved to be so incredibly similar that meanst that we're open to infection and superantigenic responses from the local environment for which we have no immunity.

Of course as this is science fiction you could tap dance past this by either not addressing the peculiar similarity with Earth or having it that the ecosystems can't mesh so humans simply clear out areas and plant terrestrial ecosystems or have it that humans use advanced immunology to give themselves immunity to an entire alien ecosystem. I've seen examples of the latter in SF before (one example that springs to mind is a story by Ken McLeod wherein the characters wear suits that take in samples from the air and soil in the environment, run the samples through sophisticated lab on chips and then inject the occupants with rapid vaccines, after a few hours they can take off their suits safely. Given that there is some progress towards genetically engineered immune systems IRL (in HIV research no less!) this isn't so much of a stretch. You could perhaps add in some gene therapy or cell therapy for digestion too such as GM or synthetic gut flora capable of breaking down alien biomolecules.

The golden rule of SF writing is that so long as you are consistent with the plot devices you throw in and don't contradict what we already know to be true then you'll be fine. Extra points for exploring the social consequences of your plot devices.
Hmm can't think of anything. In terms of dealing with it I imagine that you would get lots of tough, low-lying vegetation with widespread roots; like a rhizome mat, with flourishes of taller plants in sheltered areas like valleys or next to mountains (which incedentally may be quite tall if your planet is spinning fast). Such plants would benefit greatly from airborn seed dispersal, that and high winds might mean flying animals are rare. Animal life is likely to be heavy, short and possibly burrowing.

I'm not sure but I think the effects may be less severe at higher lattitudes so there are likely to be differences in ecosystems there, on top of that such a large planet with greater differences in climate is likely to have greater biodiversity.

BadBrain

A simple rocky body with a molten iron core (and, Ryan_m_b, a molten iron core is necessary to generate the magnetic field which would fend of the solar wind and keep it from blasting the atmosphere into outer space) is sufficient to prevent an implosion. Mars has a much smaller radius than Earth, and a similar mineral makeup, and it hasn't imploded yet. If you want to increase gravitational pull at the surface by 30% while keeping your planet's density the same as Earth's, just decrease Earth's radius by the square root of 30% of Earth's radius.

Earth's radius is 6 378.1 km, 30% of which is 1913.43 km, the square root of which is 437.4277, so 6378.1 km - 437.4277 km = 6334.3573 km.

Ryan_m_b

Of course! Don't know why I forgot that considering the implications for the ecosystem of increased radiation.

Lowkea

Excellent points BB and Ryan, I really appreciate you lending me your thoughts.
As you say Ryan good science fiction is really just good fiction, I have various plot devices in mind for my explanation of successful human habitation of this 'alien' planet. Larry Niven, Kim Stanley Robinson and Greg Bear come to mind as authors who have tread this fine line between social commentary and brain expanding fact. Consistency is the key!

qraal

BadBrain the Moon's mass is 0.0123 the Earth's and Mars's mass is 0.1075 times Earth. I think what you were trying to say is that the radius is proportional to the 1/3 root of the mass. Since the gravity is proportional to M/R² that means surface gravity is proportional to the 1/3 root of the mass too. Of course the implicit assumption is that the density is the same, but even if it's not it's just a linear proportion to the surface gravity. Thus the Moon's 3.34 g/cc density, compared to the Earth's 5.52 g/cc density means an Earth made of stuff of Lunar density would have a surface gravity of (3.34/5.52) times the strength.

I think Kepler 22b's mass is 7 Earths, but its surface gravity should be less than twice Earth's.

qraal

Lowkea a Super-earth up to 10 Earth masses should have a radius proportional to the mass with a function like so:

R = Ro.M^α

...where α is somewhere between 1/3 and 1/4. Most models of how materials shrink under compression mean an α of 0.28-0.26. Make it 0.27 for argument's sake. Ro depends on what the planet is made of - it's one Earth radius if it's made of an Earth-like mix of 2/3 rock and 1/3 iron, and it's 1.26 Earth radii if the planet is half ice, half Earth-stuff. A nearly pure rock object, like the Moon, would be a bit bigger than an 'Earth'.

Knowing that we can show that the surface gravity is:

g = M/(M^2.α) = M/(M^0.54) = M^0.46

...which means a 7 Earth mass Super-Earth has a surface gravity of 2.45 gee.

A bit of maths always helps when trying to figure these things out.

D H

You are forgetting compressibility. The density of iron at the Earth's surface is 7.9 grams/cc. The density at the Earth's core is 13.1 grams/cc -- and that's lower than that for iron (google "core density deficit").

Compressibility is a big factor here. Ignoring compressibility would lead one to think that because radius of Kepler 22b is 2.4 times that of the Earth, it's should be 13.8 (2.4³) times that of the Earth and that it's surface gravity should be 2.4 g. You cannot ignore compressibility. Kepler 22b's mass would be 40 times that of the Earth and it's surface gravity would be 7g if Kepler 22b had an Earth-like composition (which it apparently does not have).


There is no collapse. There's just compression. The iron in the Earth's core is still iron. So how can you prevent this? You can't. Solids are compressible. A planet is more or less in hydrostatic equilibrium, through and through.

You can mitigate it to some extent. Change the abundance of elements. Earth is predominantly iron (35% of the total mass), oxygen (30%) silicon (15%), and magnesium (13%). That's 93% of the Earth's mass. Replace some of that 35% iron with lighter elements. I don't know how much that will help you make your bigger planet. It will help some. If you need numbers, you could come up with a reasonable differentiated (iron at the core, denser rock in the mantle, light fluffy rock in the crust) mass distribution, set it in hydrostatic equilibrium with reasonable bulk properties, and see what you get. Add more mass. See what you get.

That reduced iron probably means a paucity of other metals as well. A metal poor planet means your natives aren't going to progress beyond stone age. No bronze age, no iron age. They might be smart, but they probably can't be technological.

phyzguy

Attached is a graph showing planetary masses and radii. The underlying graph is from Sara Seager at MIT, and gives the relation between mass and radius depending on the composition. I added the two lines of constant surface gravity. You can see there are a large range of planet sizes that can have gravities similar to Earth's. Some of these hypothetical water planets (in the blue box) can be 2 or 3 Earth masses and still have a reasonable gravity. Whether they would be habitable for other reasons (like radiation, as already mentioned) is another question.

This is a few years old - many more exoplanets are known now.