Reasonable mass and surface gravity of fictional planet based on radius

[FtlIsAwesome]

How can I give a fictional planet a realistic mass and surface gravity based on radius? Mercury and Mars are near equal in surface gravity strength, so it does seem that I have some room for variation.

From Wikipedia:
Earth
Average radius 6,371.0 km
Surface gravity 9.8 m/s²
Mass 5.9736x10²⁴ kg
Average density 5.515 g/cm³

 

[cepheid]

It sounds like you're saying you want to fix the radius and determine the mass and surface gravity accordingly. To do that, you have to decide what your planet is made out of, which gives you an idea of its average density. That will enable you to determine the total mass, and hence the surface gravity.

 

[FtlIsAwesome]

Currently I am working on various details for my scifi series.

As the size of a terrestrial planet goes up, it makes sense for the density to increase as well, and vice versa. I don't want to give the planets an extremely undervalued or overvalued density for their size.
I guess my question could be labeled as "worldbuilding tips".Here are two examples I worked out a few months ago:

Planet Ramson
Radius - about 6,250 km
Surface gravity - 0.96 g - 9.4143 m/s²
Mass - 0.9245 MEarth - 5.5226x10²⁴ kg
---Volume - 1.026x10¹² km³

Planet Armstrong
Radius - about 8,240 km
Surface gravity - 1.12 g - 10.983 m/s²
Mass - 1.872 MEarth - 1.1181x10²⁵ kg
---Volume - 2.3446x10¹² km³

I think I have my math messed up somewhere by 1,000:
---Ramson Density - 5.383x10¹² kg/km³ - 5.383x10^-3 g/cm³
It sounds about right except for that 10^-3.On this forum I am not sure as to how much detail I should go into concerning my fictional universe.

It sounds like you're saying you want to fix the radius and determine the mass and surface gravity accordingly.

Yes, this is exactly what I am trying to do.

To do that, you have to decide what your planet is made out of, which gives you an idea of its average density. That will enable you to determine the total mass, and hence the surface gravity.

Unfortunently I currently have little idea as to the composition of my worlds, other than they are terrestrial and rocky. The two I mentioned above are habitable by humans, but their composition could be different from Earth's.

 

[Nabeshin]

As the size of a terrestrial planet goes up, it makes sense for the density to increase as well, and vice versa.

Hrm. It is definitely not this simple, and this may even be downright wrong. It has much more to do with how the planet was formed than its final mass (or radius, however you want to define size). Certainly, at some point the planet begins to become gaseous, and density is undoubtedly lower there!

Unfortunently I currently have little idea as to the composition of my worlds, other than they are terrestrial and rocky. The two I mentioned above are habitable by humans, but their composition could be different from Earth's.

Well, there's really nothing that can be said here. In my opinion, it makes perfect sense to just quote the surface gravity as 1.5g or 0.5g and leave it at that. If you'd like to go further, a possible explanation for either of these values would be comparatively large/small iron cores. This also would then be linked to the magnetic field of the planet.

 

[FtlIsAwesome]

I have a 12,090 km radius terrestrial planet, which I have stabbed its gravity between 1.75 and 2.75 g. I'm leaning to the smaller value, as I don't want it to be too harsh for humans.

Is there a post concerning planets' composition?

Terrestrial planets could possibly be very large.
GJ 1214 b is a large exoplanet of ~17,000 km radius, and it could be either a jovian or a terrestrial.
Could a rocky terrestrial planet be larger than Neptune? Such a planet wouldn't be common, but I only need one out of all the millions of other existing planets. Although such a planet might obtain a thick atmosphere and end up with properties similar to a jovian.

In my opinion, it makes perfect sense to just quote the surface gravity as 1.5g or 0.5g and leave it at that.

Generally I would expect an Earth-sized planet to have 1 g, but I would find a 0.75 g Earth-size planet believable.
A large planet wouldn't be able to have a low density because the gravity would simply compress it to a smaller size.

 

[cepheid]

5.4e12 kg/km³ is correct for planet Ramson's density. To convert to g/cm³, you have to multiply by two conversion factors, the first being (10³ g/kg) and the second being (1 km³ / 10¹⁵ cm³). This gives you an overall factor of 10^-12, which cancels with the 10¹² to leave you with 5.4 g/cm³.

The second conversion factor comes from this: 1 km³ = (10³ m)³ = (10⁵ cm)³ = 10¹⁵ cm³

The density is really a property of the material, so if you want to develop all of the physical properties of your planet for the story, I would say that composition/density would be the starting point. On the other hand, Nabeshin's suggestion of just stating the surface gravity and moving on might be the way to go, unless the other physical characteristics of the planet are actually story points.

 

[FtlIsAwesome]

The density is really a property of the material, so if you want to develop all of the physical properties of your planet for the story, I would say that composition/density would be the starting point. On the other hand, Nabeshin's suggestion of just stating the surface gravity and moving on might be the way to go, unless the other physical characteristics of the planet are actually story points.

In the series I probably won't state the density of the planets, but the density would determine gravity which is something I would say in the story. The gravity would affect jumping, mobility, spacelaunches, the orbital periods of artificial satellites and natural moons, etc.
I don't want to stick any old gravity on a planet, rather I want to start with a realistic density and get the surface gravity and mass from that.

I probably should get around to determining the planets' internal composition. Any tips on that?

 

[Nabeshin]

I probably should get around to determining the planets' internal composition. Any tips on that?

This is exactly what I was trying to get at with my earlier post. You have so much freedom to choose the internal composition and still have things be scientifically plausible. Obviously the planet is not uniform in density, so you have to decide what kind of layering you want (just core/mantle or more layers than that?). And of course, you would need to decide how big these individual shells are and what they are composed of.

The net result is a large number of different interiors which yield the same surface gravity. So really, if you're not going to either a) describe the planetary interior in great detail, or b) invoke the interior for some further planetary description (i.e. magnetic field, plate tectonics, I don't know, something), then I just don't see how it could possibly be beneficial to choose one out of the infinite schemes which will give you a surface gravity of, say, 1.08g.

 

[FtlIsAwesome]

Can size/mass affect composition and if so how?

 

[qraal]

Can size/mass affect composition and if so how?

It can and does. Here's how. Earth and Venus, for example, have densities over ~5, but their uncompressed densities are about 4.2. Pressure causes all materials to decrease in volume and so the larger the planet, the larger the core pressure and the larger the compression of its constituents. Earth's core, for example, varies in pressure from about 1.5 million bar (i.e. 150 GPa) to about 3 million bar (300 GPa) and the iron/nickel alloy it's composed of would have a density of about 7, but in the Core has an average density of 11.

There are a number of papers online which cover the smaller planet range, with a variety of compositions. Have a look around arxiv.org for papers on "mass-radius relationships" for planets and you'll find quite a few. A good rough estimate for planets like Earth (i.e. 75% rock and 25% iron/nickel) is...

R(M) = Ro*M^b

with b in the range 0.272-0.268 for planets massing between 1 - 10 Earth masses, and Ro is 1 Earth radius. For planets smaller than Earth, then b ~ 0.3 will do. For planets that are ~50:50 Ice : Rock/Metal, then Ro ~1.26. Basically a super-Ganymede or Callisto.

For planets like Mercury (25% rock and 75% iron/nickel) the radius relation for the range 1-10 Mercury masses has Ro = Mercury's radius and b = 0.3.

Surprisingly neither the Moon nor Mars have Earth-like interiors. Both contain more silicates than the Earth does. Mars's uncompressed density is ~3.78 vs 3.93 compressed. The Moon's is ~3.32 vs 3.344 compressed.

Past 10 Earth masses and things get complicated. That's the mass that a planet can start sucking excess gas from the proto-solar nebula. A little bit of gas and you get Uranus or Neptune. A lot more gas and you get Jupiter and Saturn. Jupiter is very close to the maximum size of a planet made of Hydrogen/Helium - beyond about 3.24 Jupiter masses and the radius of a planet is smaller than Jupiter's because its core is composed of degenerate matter, causing the planet to shrink as the mass is increased.

Because of this Brown Dwarfs are all about the same size as Jupiter, unless there's a large internal release of heat (from gravitational collapse, tides, and ohmic heating) to "inflate" the planet's outer layers, as appears to oocur on a lot of exosolar Transiting Gas Giants. A brown dwarf that masses ~50 times Jupiter is actually just a bit smaller after ~5 billion years of cooling down, and so has a density ~70 times greater than water.

Stars reverse the shrinking trend because of fusion energy being released in their cores. When that runs out, they shrink into White Dwarfs roughly the size of Earth, but massing as much as the Sun - thus over 1,000,000 times denser than water.

 

[FtlIsAwesome]

I tried using your equation but I couldn't get the numbers right.

I searched arxiv.org and found this:
http://arxiv.org/PS_cache/arxiv/pdf/0912/0912.2968v1.pdf
The section "4.2. Transiting extrasolar planets" has some useful equations for terrestrial worlds.R = R_ref(M/M_⊕)^β
R_ref = (1 + 0.56α)R_⊕

β = 0.262(1 − 0.138α) for rocky/ocean superEarths
or
β = 0.3 for planets between 10⁻² to 1 M_⊕

α is the fraction of the planet's mass that is water, a large value lowers the mass of the planetUsing these equations my 12,090 km planet has a mass of 11.5 M-Earth, assuming α=0.001. Slightly surprising, it has a surface gravity less than Earth's, 0.3 g. Its density is 9.3 g/cm³, so I'm on the right track. Because of its high mass, it might have a thick atmosphere.

I theorize that there is an intermediate class of planets between terrestrials and jovians. These planets have large cores that are close in mass to their atmosphere. They could be called the "superVenus" class, though they would have several differences from Venus. My 12,090 km planet above will be just under the intermediate class.

This paper
http://arxiv.org/PS_cache/arxiv/pdf/1003/1003.0451v1.pdf
seems to have some information that might be useful, but I only skimmed it.

Is there a post concerning planets' composition?

Oops. I meant thread not post.

 

[FtlIsAwesome]

it has a surface gravity less than Earth's, 0.3 g

Math Error. Its actual gravity is 31.3 m/s², or 3.2 g.

 

[redwood973]

I've sent a private message or two FtlIsAwesome's way regarding this topic, and we're curious as to your opinions. Here is the original message I sent him--

Hello, I was going through threads after discovering this forum and came upon your “Reasonable mass and surface gravity of fictional planet based on radius.”

There has been no further activity on the thread, so maybe assistance/input is no longer necessary but I’ve had some experience with role playing games which may shed some light on the topic for you if you are still in need.

I have seen two ways of doing this–

[From Steve Jackson's GURPS: Space (2nd Edition)]

diameter x density x 0.0000285 = gravity

For Earth this would be 6,371 x 5.515 x 0.0000285 for a gravity of 1.00. As long as you have two of the components you can easily define the third; for example your planet Armstrong, if we only knew it’s gravity was 1.12 and its radius is 8,240 we have: 1.12 / 0.0000285 / 8,240 and get a density of 4.769.

The second way I’ve seen–

[From a home-brew system, to which I cannot recall the source. . .]

Mass (m) = (radius (r) / 6,371)³ * density compared to Earth

Gravity (g) = m / (r / 6,371)²

So, using Planet Armstrong we have (8,240 / 6,371)³ * 0.864 for a mass of 1.869 (compared to Earths). Continuing, we have 1.896 / (8,240 / 6,371)² for a gravity of 1.13 which due to the freakish nature of rounding is 0.01 g stronger than the established 1.12 g from the first option.

The problem with this second option is needing to know the density of the body in question compared to Earth.

I hope you find this helpful in some manner.

So, thoughts and comments?

 

[qraal]

I've sent a private message or two FtlIsAwesome's way regarding this topic, and we're curious as to your opinions. Here is the original message I sent him--... snipped ...So, thoughts and comments?

As good as any rough estimate rule. The joy of planet-making lies in the degree you can justify your numbers and compare to real planets.

 

[redwood973]

As good as any rough estimate rule. The joy of planet-making lies in the degree you can justify your numbers and compare to real planets.

Ooooooh, project time! Be right back. . .

 

[redwood973]

Okay, after a quick format of a spreadsheet and a little information from the wikipedia I came up with the following--

Planet/Actual Gravity/Gravity acording to GURPS/Gravity acording to home-brew
Mecury/0.38/0.38/0.38
Venus/0.90/0.90/0.89
Earth/1.0/1.0/1.0
Mars/0.38/0.38/0.38
Jupiter/2.53/2.7/2.69
Saturn/1.07/1.18/1.17
Uranus/0.89/0.93/0.92
Neptune/1.14/1.16/1.15

So, it seems both equation work well for terrestrial worlds, but are a little off for jovian worlds (Jupiter off by an average of 0.065; Saturn by 0.095; Urauns by 0.035; and Neptune by 0.015). The larger the world, the less accurate the result--hrm, although Juipiter's results are closer to fact than Saturn's.