# What's the largest an Earth-like planet can be and support Earth's biosphere?

For certain reasons I decided to not set my story on Earth. However, the planet is meant to host an Earth-like biosphere (including humans, most of Earth's species (perhaps some that didn't evolve before)). Since I was working with another planet, I decided to make it as big as possible (hence the question). The lifeforms on said planet don't need to be exactly analogous to Earth (evolution could have taken different forms), but it does need to be able to support homo sapiens (with perhaps some biological adaptations to living under higher gravity (only as high as is feasible) and other differing conditions, but still the same basic makeup).

Surface gravity is directly proportional with the radius of the planet and inversely proportional with the density of the planet.

Now fire up LibreOffice Calc (or your favorite spreadsheet program), and play with the numbers. I suggest to put Earth's radius, surface area and surface gravity as 1 (because your are interested in relative values), but keep the Earth's density as 5.5 (because it has direct implications for the chemical composition of the planet etc.) Then figure out what increase in surface gravity you are prepared to tolerate (I suggest 20 to 25% tops) and what decrease in density you can justify while keeping enough iron to get a decent magnetic field and to preserve Earth-like biochemistry (I suggest not lower than 5). You will get something like this:

                                Radius  Area  Density Gravity
------  ----  ------- -------
Earth                             1.00  1.00     5.50    1.00
Max gravity, lowest density       1.38  1.89     5.00    1.25
Max-ish gravity, lowest density   1.32  1.74     5.00    1.20
Moderate gravity, lowest density  1.21  1.46     5.00    1.10


This suggests that you can get a surface area 75 to 90% larger than Earth's without extremely strong effects on the biosphere, and a surface area 50% larger than Earth's with minimal effects on the biosphere.

• Thanks. I'll probably go with the 50% larger since I'm not sure what the limitations of "without extremely strong effects on the biosphere" entail. – Tobi Alafin Jul 20 '19 at 16:16
• @AlexP how big is 50% larger? – Incognito Jul 20 '19 at 16:31
• @Incognito: Surface area one and a half times as large as Earth's? I don't understand the question. – AlexP Jul 20 '19 at 16:33

We don't have a good measure for how much gravity a human being can sustain for a lifetime. We might well find that 1.1 g is too much. Or that we can easily adapt to 2g. We have a little better understanding of microgravity's effects on the human body, but almost none on long term higher gravity. Twice is probably too high, but anywhere from 1.1 to 1.5 is up to you. No one can tell you that it is wrong, because we just don't know.

All our studies of higher gravity are based on limited duration. Basically the length of the high acceleration trip. We can't maintain high acceleration for a long period of time (too energy intensive), so we don't know what the effects are.

Beyond that, bigger doesn't necessarily mean higher gravity. If the planet is less dense (for example, no iron core), it can have a higher volume/surface area and the same gravity. The formula is

$$g = \frac{Gm}{r^2}$$

The formula for mass is volume times density.

$$m = \frac{4\pi r^3}{3}\rho$$

Substituting, we get

$$g = \frac{4G\pi r\rho}{3}$$

Rearranging

$$r = \frac{3g}{4G\pi\rho}$$

$$G$$ and $$\pi$$ are constants. Now, let's rewrite this as a proportion.

$$\frac{r}{r_E} = \frac{g{\rho}_E}{g_E\rho}$$

What this says is that the size of the radius in Earth radii is equal to the number of Earth gravities divided by the density in Earth densities. So taking out the proportions, we have

$$r = \frac{g}{\rho}$$

There are some limits to how low you can make the density. Jupiter is composed of things (e.g. hydrogen) that have a low density at Earth gravity. But because Jupiter is so massive, they are compressed to a much higher density.

Mars' density is a bit more than three quarters that of Earth. All the lower density planets in our solar system are gas giants. So you can probably get three quarters. Putting that back into our formula, we get

$$r = \frac{1.5}{.75} = 2$$

So about the most you can expect to get is a planet twice as large in radius as the Earth. This will also have four times the surface area and eight times the volume. And six times the mass.

It is possible that you can manage a lower density than Mars. I don't have a good way of evaluating it. If you find that you can, you can put that number back into the formula.

• Can a planet with Mars' like density support the biosphere? Apart from gravity, there's solar radiation? Wouldn't we still need a strong magnetic field? – Tobi Alafin Jul 20 '19 at 16:35

Steven Dole suggested in his book Habitable Planets for Man the following gravity, mass and radius ranges for planets still maintaining a magnetosphere, plate tectonics, and a nitrogen + oxygen atmosphere. All values will be given relative to Earth.

$$M = 0.4 - 2.35$$

$$R = 0.78 - 1.25$$

$$g = 0.68 - 1.5$$

Now you want a big planet, yet you do not specify what you mean with big... Thus I'll calculate several examples.

$$M = 2.35$$

$$R = 1.25$$

$$g = \frac{M}{R^2} = \frac{2.35}{1.25^2} = 1.5$$

$$A = 4*\pi*R^2 = 4*\pi*1.25^2 = 1.56$$

$$M = 2.35$$

$$R = 0.78$$

$$g = \frac{M}{R^2} = \frac{2.35}{0.78^2} = 0.25$$

$$A = 4*\pi*R^2 = 4*\pi*0.78^2 = 0.61$$

$$M = 0.4$$

$$R = 1.25$$

$$g = \frac{M}{R^2} = \frac{0.4}{1.25^2} = 0.25$$

$$A = 4*\pi*R^2 = 4*\pi*1.25^2 = 1.56$$

However, there is an issue I see concerning the Max_Max case. It is called atmospheric escape and can ruin your day during planet formation. You can see that whether or not a gas will remain in the planet's atmosphere depends on the escape velocity given by

$$v_{esc} = \sqrt{\frac{M}{R}}$$

and the temperature of a planet is given by

$$T_{eq} = T_{star}*(1-Ab) ^\frac{1}{4}*\sqrt{\frac{R}{2a}}$$

$$Ab = \text{albedo}$$

$$a = \text{distance}$$

From this I get $$v_{esc}$$ of 15.34 km/s for the Max_Max scenario, which gets awfully close to the point where it retains helium and would turn into an ice or gas-giant. Atmospheric escape is the make it or break it point in the end for the survival of an Earth-like biosphere. You need methane, ammonia and water to stay on the planet and you need helium and hydrogen to leave. Otherwise it is utterly impossible for an earth-like biosphere to be sustained.

• Thanks for the link to the article, I'll check it out. – Tobi Alafin Jul 21 '19 at 13:55

According to my rough calculations, a planet habitable for water based lifeforms vaguely similar to terrestrial life - not guaranteed to be habitable for humans or other lifeforms transported from Earth - could have a surface area a little more than 1.5 times that of Earth, which is rather disappointing. Some other answers also support that surface area limit through other calculations.

(added 07-26-19. But these calculations of upper limits are still rather uncertain and controversial.)

I believe that the habitablility of a planet of a given size depends a lot on its distances from its star and how much heat and light it gets from its star, so that larger planets would be more likely to be habitable farther out from their stars, and on various other factors.

Possibly an expert on planetary science and astrobiology could calculate and design an alien planet with a significantly larger surface area, with a larger percentage of ocean or dry land as you may prefer, and habitable for humans and other Earth life forms.

You might want to consider where you want your story to be on the MOHS Scale of Science Fiction Hardness.

The harder - more realistic and plausible - you want your science fiction story to be, the more the size of your planet will be constrained by various scientific factors.

Many old fashioned science fiction stories imagined that the giant planets in our Solar System and similar sized exoplanets could have solid surfaces and biospheres. Thus they depicted habitable planets with tens, hundreds, and thousands of times the surface area of Earth.

In E.E. Smith's Lensman series the heavy gravity planet Valeria is settled by Earth Humans and centuries or millennia later their descendants have adapted and are immensely strong. I forget what the surface gravity of Valeria was but it was probably far higher than humans could actually survive in.

As I remember from checking fairly recently, Stephen Dole's Habitable Planets for Man (1964, 2009) suggests that humans wouldn't want to colonize a planet with a surface gravity more than about 1.25 or 1.50 that of Earth. The surface gravity of Earth is abbreviated 1 g.

A writer could get away with having a group of colonists or alien abductees settle on a planet with a surface gravity of 1.10 g, and then after generations of adaptation have a group of their descendants settle on a planet with a surface gravity of 1.21 g. By repeating this process over and over again over generations, centuries, and millennia, planets with surface gravities of 1.331 g, 1.4641 g, 1.61051 g, 1.771561 g, etc., can be settled until eventually some absolute upper limit is reached.

Or possibly genetic engineering could be used to modify Earth Humans to be able to survive, be healthy, and function on planets with higher gravity than Earth. If it is a fantasy story some type of magic could modify Earth Humans to live on the planet.

Or maybe the natives of that planet aren't Earth Humans but members of another species that look a lot like humans, except probably being shorter and stockier. And if there aren't any characters from Earth in the story the characters would mostly be described by how they appear to other members of their species and there might not be more than a few subtle clues as to how different from Earth Humans they are.

Another factor to consider is plate tectonics, which are considered to be a factor in making Earth habitable. Many smaller astronomical bodies in our Solar System don't have plate tectonics. So one would think that a planet larger than Earth wouldn't have any problems with insufficient plate tectonics.

But there is an article:

"Exomoon Habitability Constrained by Illumination and Tidal heating" by Rene Heller and Roy Barnes, Astrobiology, January 2013.

In section 2, Habitability of Exomoons, they discuss the mass range necessary for hypothetical exomoons to be habitable in the sixth paragraph:

A minimum mass of an exomoon is required to drive a magnetic shield on a billion-year timescale (Ms≳0.1M⊕; Tachinami et al., 2011); to sustain a substantial, long-lived atmosphere (Ms≳0.12M⊕; Williams et al., 1997; Kaltenegger, 2000); and to drive tectonic activity (Ms≳0.23M⊕; Williams et al., 1997), which is necessary to maintain plate tectonics and to support the carbon-silicate cycle. Weak internal dynamos have been detected in Mercury and Ganymede (Gurnett et al., 1996; Kivelson et al., 1996), suggesting that satellite masses>0.25M⊕ will be adequate for considerations of exomoon habitability. This lower limit, however, is not a fixed number. Further sources of energy—such as radiogenic and tidal heating, and the effect of a moon's composition and structure—can alter the limit in either direction. An upper mass limit is given by the fact that increasing mass leads to high pressures in the planet's interior, which will increase the mantle viscosity and depress heat transfer throughout the mantle as well as in the core. Above a critical mass, the dynamo is strongly suppressed and becomes too weak to generate a magnetic field or sustain plate tectonics. This maximum mass can be placed around 2M⊕ (Gaidos et al., 2010; Noack and Breuer, 2011; Stamenković et al., 2011). Summing up these conditions, we expect approximately Earth-mass moons to be habitable, and these objects could be detectable with the newly started Hunt for Exomoons with Kepler (HEK) project (Kipping et al., 2012).

The upper limit of about 2 times the mass of Earth should hold for exoplanets as well as exomoons.

Heller and Barnes give the source for the importance of plate tectonics for habitability as:

Williams D.M. Kasting J.F. Wade R.A. Habitable moons around extrasolar giant planets. Nature. 1997;385:234–236. [PubMed] [Google Scholar]

Heller and Barnes give the sources for an upper mass limit at about 2 Earth masses as:

Gaidos E. Conrad C.P. Manga M. Hernlund J. Thermodynamics limits on magnetodynamos in rocky exoplanets. Astrophys J. 2010;718:596–609. [Google Scholar]

Noack L. Breuer D. Plate tectonics on Earth-like planets [EPSC-DPS2011-890]. EPSC-DPS Joint Meeting 2011, European Planetary Science Congress and Division for Planetary Sciences of the American Astronomical Society; 2011. [Google Scholar]

Stamenković V. Breuer D. Spohn T. Thermal and transport properties of mantle rock at high pressure: applications to super-Earths. Icarus. 2011;216:572–596. [Google Scholar]

It is possible that the importance of plate tectonics for habitability, and the upper mass limit of about two times the mass of Earth for plate tectonics, are not accepted by all scientists interested in astrobiology, but I have not researched that.

Accepting for the moment that about two times the mass of Earth is an approximate upper limit for planetary plate tectonics and planetary habitability for native lifeforms, the diameter, and thus surface area, of a planet is not solely determined by its mass. The diameter and surface area of a planet is determined by its mass and its overall density.

The overall density of a planet is determined by two factors.

One factor is the normal density of the various elements, compounds, and mixtures that it is made of, averaged. The normal density of those materials is the same density that they have floating around in tiny meteoroids in outer space, or lying on the surface of planets.

The other factor is the degree to which those materials are compressed by vast pressures at various levels in the interior of the planet, thus becoming denser and increasing the overall density of the planet.

Since the cube root of two is approximately 1.25992, a planet with twice the mass of Earth and the same overall density would have about 1.25992 times the radius and diameter of Earth and about 1.5873 times the surface area. Note that in order to have the same overall density as Earth, a planet with twice the mass of Earth would have to have a different composition than Earth, affecting life on the surface in various ways.

Since the strength of gravity depends on the mass and the square of the distance, the surface gravity of a planet with twice the mass of Earth and 1.25992 times the radius would be about 2 divided by 1.5873984, or about 1.2599231 that of Earth.

By decreasing the overall density of your planet you can increase its surface area but there is no doubt a limit to how much you can do so while keeping the planet habitable, or even with a solid surface.

A rapidly rotating planet would be more likely to have an internal dynamo driving plate tectonics, and a rapidly rotating planet would be more oblate, having a somewhat larger surface area and lower gravity at the equator. Earth original rotation rate was slowed down by tidal interactions with the Moon. As far as I know there is a controversy whether a large moon, which would slow down the rotation rate of a planet, is necessary for a planet to be habitable.

And if a writer needs a really vast world, they could set their story on an artificial habitat in space, that has a much greater surface area than any habitable planet, created by a very advanced civilization.