Two planets, vastly different sizes, similar surface gravities.

[SMJB]

Hi. First post here. I'm writing a sci-fi novel, and the above is what I need. Thanks in advance for your help.

By "similar surface gravity" I mean that someone from the smaller world could move to the heavier world with a minimum of health risks (they're going to have enough trouble with the local biosphere). By "vastly different sizes" I mean that I want the differential to be as vast as possible while keeping the above in mind and the fact that I want the larger world to still be a terrestrial world with continents similar in size to Earth's (and more of them).

The characters are human and the smaller world was terraformed. It has a minimal amount of water, but it's spread out rather strategically, so it's not a desert world. There was an interstellar civilization once upon a time, but it fell and cast these worlds into a new bronze age, and the smaller has risen to be approximately modern while the highest civilization on the larger is about 19th century; basically, there will be no phlebotinum-based solutions to their problems, so if the smaller world can't colonize the larger one, I'm out of a story.

 

[SMJB]

Hey, what if at some point in deep time the smaller world was larger and had it's crust and outer mantle ripped off by some collision or something, thus giving it, for its size, an unusually large (and maybe dense) iron core? Is that, you know, a thing that could feasibly happen without throwing this world into an orbit that humanity wouldn't like or out of the system altogether? This might also help with the fact that this world is supposed to have a large number of moons.

 

[snorkack]

An example from solar system:
Mercury - diametre 4880 km, surface gravity 0,38 g
Mars - diametre 6779 km, surface gravity 0,376 g

 

[Bandersnatch]

Hey, what if at some point in deep time the smaller world was larger and had it's crust and outer mantle ripped off by some collision or something, thus giving it, for its size, an unusually large (and maybe dense) iron core? Is that, you know, a thing that could feasibly happen without throwing this world into an orbit that humanity wouldn't like or out of the system altogether? This might also help with the fact that this world is supposed to have a large number of moons.

You've just described the dominant hypothesis regarding the formation of the Earth-Moon system. It is thought that a collision of proto-Earth with a Mars-sized body in the early days of the Solar system resulted in the outer layers of both planets to be ejected, later on to coalesce into the Moon, while the denser inner parts merged to form Earth. This is to explain the significant density difference between the two bodies(about 3 vs 5.5 g/cm^3).
http://en.wikipedia.org/wiki/Giant_impact_hypothesis

For your planet, just keep in mind this simple(also, simplified) relationship:
$g=ρR$
where g is the acceleration due to gravity on the planet's surface, ρ is the average density, and R is the radius. All being expressed as fractions of Earth's(so, for Earth it's all 1s).
You want g to be the same as we've got here, so g=1. Now all you need to make sure, is that you reduce the density by the same factor as you increase the radius to remain physically plausible.

One more thing to remember is that having twice the radius means the surface area increases four times, so for most intents and purposes doubling the radius translates to quadrupling the "size" as the inhabitants would perceive it.

 

[SMJB]

An example from solar system:
Mercury - diametre 4880 km, surface gravity 0,38 g
Mars - diametre 6779 km, surface gravity 0,376 g

I looked it up, and Mercury is the perfect model for what I want. Not only is that mechanism I proposed sound, it apparently stripped off far more of Mercury's original mantle than I'd have ever dared hope was plausible.

Right, so let's scale it up to a world with a mean radius of, say, 4800 km. This world world realistically have a mean density much higher than Earth's, being that Mercury's density is already nearly the same as Earths, but if there's an equation for determining the density of such a world I don't know it, so I'm just going to say that according to this, a world with this radius and an earthlike density of 5.5 would have a gravity of .75 g, and a totally hypothetical, based-on-nothing-but-guesswork density of 6 g/cm means a gravity of .82 g. And that...that would be just about perfect. But as I said, that number is based on nothing.

 

[SMJB]

Right, I did some number-quackery assuming an average core density of 12 and an average mantle-and-crust-density of whatever the same assumption about the core would yield on Earth, giving me this equation:

((5.5*100-30*12)/70*58+12*42)/100

Like I said, it's quackery rather than science, but it's better than nothing--and it gave me a density of > 6.6! Which is .91 g! This number makes me very happy indeed.

 

[SMJB]

I've run the numbers backwards on the larger world. Assuming a radius of 9000 km (twice the surface area of Earth ought to be enough for my purposes :P) and a gravity of 1.2 g gives me an average density of 4.65 g/cc, which when run through the above equation gives me a planet that's 20% core. This seems eminently reasonable to me; it appears this star system is lower in metal than ours, but I was kind of figured that'd end up being the case, anyway. Also, being in objective numbers nearly twice the size of Earth's core means that there ought to be nothing wrong with its Van Allen belts.

Of course, this is all still quackery, but that's why I'm here; to see if my quackery stands up to scrutiny. Also to see what consequences my decisions have for my worlds that I might not expect--for instance, all those ridges on Mercury from a late-cooling core. I'd figure there'd be less of them on my smaller world, being that everything's bigger there (slower cooling all around, and higher gravity means shorter mountains).

 

[CraigDxHypo]

How much is "vastly different"?

Two planets, vastly different sizes, similar surface gravities.
…
By "vastly different sizes" I mean that I want the differential to be as vast as possible while keeping the above in mind and the fact that I want the larger world to still be a terrestrial world with continents similar in size to Earth's (and more of them).

If a difference in ratio of radii of 2 (and thus a surface area ratio of 4) satisfies the “vastly different” requirement, this is pretty easy to imagine:

Start with two roughly Earth-like planets (let’s call them “UE” and “LE” for “usual” and “little” Earth), and have one undergo an impact with a Mar-like planet, but rather than an indirect impact like the one theorized to have resulted in the formation of the Moon, a direct, more destructive one. Have LE much closer to the Sun than usual, so that it’s lower density, mostly silicate bits, are blown away by the solar wind, resulting in mostly iron remnant that re-coalesces into a “core only” version (with enough silicates to form an Earth-like crust, so perhaps best to call a “mantle-less” version) of UE, with about 1/3rd the mass and 1/2 the radius and 8/9ths (1/3 / (1/2)^3) the surface gravity.

The precise calculations for all this are tricky, because they must account for differences in pressure and temperature between LE and UE. The room temperature and pressure density of iron and nickel are 7.87 and 8.91 g/cm^3, the usual core 95/5% mixture, 7.93, but the density of UE’s core varies from about 9.9 (1.25 times) at the outermost of its liquid outer part to 13.1 (1.65) in its solid inner part. Without the pressure caused by its missing outer parts, LE’s “naked core” will have a slightly greater radius and significantly lesser density. There’s no need to sweat the details (other than for fun :) ), though, since we can imagine the pre-giant collision LE as being just enough more massive than UE to compensate for this.

A tricky part in this imagination scenario is how to get LE from the close-to-the-Sun orbit needed to blow away its silicates after its giant impact to a comfortable Earth-like one. A complicated scenario involving a “wandering” giant planet with a very eccentric orbit that “ejected” LE from its close orbit, then luckily circularized it comes to mind. Nothing like that is theorized to have happened in our solar system, and the likelihood of it by a plausible solar system formation model is, I’d WAG, very small, but there are a truly vast number of Sun-like solar systems in the universe, so “unlikely” isn’t a barrier to reasonable imagination.

Another tricky bit of imagining is where to put the two Earths, UE and LE. To be comfortable – that is, to have liquid surface water and similar atmospheres, they’d need to be in about the same orbit, but planets these sizes in the similar orbits is nearly a dynamic impossibility. If their orbits were exactly the same, they could just be 180% opposite, but such exactitude is vastly unlikely. If their orbits are only similar, eventually they’ll interact, colliding, or changing their orbits into something Earth-Venus or Earth-Mars-like, making one world hotter, the other colder.

A UE/LE radius ratio of 2 isn’t “vast” enough – if, for example, your plot need a radius ratio of 10 (surface area ration of 100), is harder for me to imaging. Making LE’s core out of a much denser elements (gold and platinum, for example, have densities 19.282 and 21.46 g/cm^3, vs iron-nickel’s 7.93), while keeping UE’s the usual iron-nickel, for example, could get you a radius ratio of about 5 (surface area 25 times). To get a radius ratio greater than 5, using this kind of scenario, some implausibly exotic material would be needed.

I think it’s a good idea to reconsider one of your story-building assumptions, SMJB:

By "similar surface gravity" I mean that someone from the smaller world could move to the heavier world with a minimum of health risks

Assuming the person in question is healthy and genetically earthling-like, I don’t think moving from a lower gravity world, such as the Moon, to one like Earth, would much of a long term-health risk. An adjustment period would be needed, similar to what one needs when recovering from a period of long-inactivity, such as severe multiple fractures or a long coma, but a body evolved to living in 1 g acceleration should eventually be OK in it, regardless of personal history. We don’t at present know with certainty, but I strongly expect that an ordinary human wouldn’t have health problems living for a long time with accelerations as low as 0.1 g. So, as far as gravity goes, your LE and UE could simply as like the Moon and Earth.

More critical than its direct health impact on people, for both worlds to have earthling-friendly environments, gravity on LE and UE must retain a breathable atmosphere. They don’t have to be especially close to one another in pressure - about 0.07 times Earth’s is enough to keep body fluids from boiling, while pressures of up to about 50 times are livable if it doesn’t have too many gasses toxic at that pressure (such as nitrogen). Staying in this range isn’t easy, though, as the 2 most-Earth-like planets in or solar system show. Mars, at 0.107 Earth’s mass, and 0.378 its gravity, has only 0.006 ATMs pressure. Venus, 0.815 Earth mass 0.908, has 92 ATMs.

The great explaining trick in having 2 much different size earthling-friendly worlds isn’t, I think, having comfortable surface gravities, but having breathable atmospheres.

 

[SMJB]

If a difference in ratio of radii of 2 (and thus a surface area ratio of 4) satisfies the “vastly different” requirement, this is pretty easy to imagine:

Start with two roughly Earth-like planets (let’s call them “UE” and “LE” for “usual” and “little” Earth), and have one undergo an impact with a Mar-like planet, but rather than an indirect impact like the one theorized to have resulted in the formation of the Moon, a direct, more destructive one. Have LE much closer to the Sun than usual, so that it’s lower density, mostly silicate bits, are blown away by the solar wind, resulting in mostly iron remnant that re-coalesces into a “core only” version (with enough silicates to form an Earth-like crust, so perhaps best to call a “mantle-less” version) of UE, with about 1/3rd the mass and 1/2 the radius and 8/9ths (1/3 / (1/2)^3) the surface gravity.

The precise calculations for all this are tricky, because they must account for differences in pressure and temperature between LE and UE. The room temperature and pressure density of iron and nickel are 7.87 and 8.91 g/cm^3, the usual core 95/5% mixture, 7.93, but the density of UE’s core varies from about 9.9 (1.25 times) at the outermost of its liquid outer part to 13.1 (1.65) in its solid inner part. Without the pressure caused by its missing outer parts, LE’s “naked core” will have a slightly greater radius and significantly lesser density. There’s no need to sweat the details (other than for fun :) ), though, since we can imagine the pre-giant collision LE as being just enough more massive than UE to compensate for this.

A tricky part in this imagination scenario is how to get LE from the close-to-the-Sun orbit needed to blow away its silicates after its giant impact to a comfortable Earth-like one. A complicated scenario involving a “wandering” giant planet with a very eccentric orbit that “ejected” LE from its close orbit, then luckily circularized it comes to mind. Nothing like that is theorized to have happened in our solar system, and the likelihood of it by a plausible solar system formation model is, I’d WAG, very small, but there are a truly vast number of Sun-like solar systems in the universe, so “unlikely” isn’t a barrier to reasonable imagination.

So basically, the Mercury model I hypothesized earlier, only it turns out there's a problem because the planet once formed would need to move to a better orbit. I'm open to suggestions on that front.

Another tricky bit of imagining is where to put the two Earths, UE and LE. To be comfortable – that is, to have liquid surface water and similar atmospheres, they’d need to be in about the same orbit, but planets these sizes in the similar orbits is nearly a dynamic impossibility. If their orbits were exactly the same, they could just be 180% opposite, but such exactitude is vastly unlikely. If their orbits are only similar, eventually they’ll interact, colliding, or changing their orbits into something Earth-Venus or Earth-Mars-like, making one world hotter, the other colder.

Oh, well, they're actually orbiting two separate components of a double-star system.

A UE/LE radius ratio of 2 isn’t “vast” enough – if, for example, your plot need a radius ratio of 10 (surface area ration of 100), is harder for me to imaging. Making LE’s core out of a much denser elements (gold and platinum, for example, have densities 19.282 and 21.46 g/cm^3, vs iron-nickel’s 7.93), while keeping UE’s the usual iron-nickel, for example, could get you a radius ratio of about 5 (surface area 25 times). To get a radius ratio greater than 5, using this kind of scenario, some implausibly exotic material would be needed.

The numbers I listed upthread would be pretty much ideal.

I think it’s a good idea to reconsider one of your story-building assumptions, SMJB:Assuming the person in question is healthy and genetically earthling-like, I don’t think moving from a lower gravity world, such as the Moon, to one like Earth, would much of a long term-health risk. An adjustment period would be needed, similar to what one needs when recovering from a period of long-inactivity, such as severe multiple fractures or a long coma, but a body evolved to living in 1 g acceleration should eventually be OK in it, regardless of personal history. We don’t at present know with certainty, but I strongly expect that an ordinary human wouldn’t have health problems living for a long time with accelerations as low as 0.1 g. So, as far as gravity goes, your LE and UE could simply as like the Moon and Earth.

Maybe, but I'd rather make as few assumptions as I can get away with.

More critical than its direct health impact on people, for both worlds to have earthling-friendly environments, gravity on LE and UE must retain a breathable atmosphere. They don’t have to be especially close to one another in pressure - about 0.07 times Earth’s is enough to keep body fluids from boiling, while pressures of up to about 50 times are livable if it doesn’t have too many gasses toxic at that pressure (such as nitrogen). Staying in this range isn’t easy, though, as the 2 most-Earth-like planets in or solar system show. Mars, at 0.107 Earth’s mass, and 0.378 its gravity, has only 0.006 ATMs pressure. Venus, 0.815 Earth mass 0.908, has 92 ATMs.

The great explaining trick in having 2 much different size earthling-friendly worlds isn’t, I think, having comfortable surface gravities, but having breathable atmospheres.

Well, one of the worlds is terraformed, so there isn't a whole lot of trouble there, at least. I'd like to give the other one an atmosphere that's earthlike in composition and pressure, as well, as otherwise I'd worry about it ripping up the colonists' lungs.