(This is an expanded version of an earlier post, taking into account escape velocity as well as surface gravity.)
Earth-sized exoplanets have been very much in the news lately. In early November astronomers reported the discovery of HD 40307g, a rocky planet orbiting in its star’s habitable zone. (Centauri Dreams wrote up a good discussion of it here.) It joins a list of six other planets thought to be in the same situation.
But it’s way bigger than Earth. The Habitable Exoplanets Catalog estimates its mass as 8.2 times that of Earth. Doesn’t that mean it would have a crushing gravity? Wouldn’t a petite 100-pound woman find herself lugging around a weight of 820 pounds? Gravity might make it impossible to get off the planet. A civilization would never become spacefaring.
That’s what I thought until Abel Mendez, who runs the Habitable Exoplanets Catalog, told me the formula for calculating a planet’s surface gravity: mass divided by the radius squared. That is, SG=M/R^2.
Let’s try it with HD 40307g, using data from the Habitable Exoplanet Catalog. Mass, 8.2 Earths. Radius, 2.4 times that of Earth. That gets you a surface gravity of 1.42 times Earth.
It seems counterintuitive, doesn’t it? How can a planet be so much more massive than Earth yet have only 1.42 times the gravity at the surface? The answer lies in the radius. The further you are from the planet’s center, the less its gravity pulls at you. Another way of putting it is that the greater the planet’s radius is for its mass, the less dense it is.
Let’s test that. Jupiter has 317.8 times the mass of Earth. That’s a lot. Yet its radius is also 11.2 times larger. Do the math and you get a surface gravity of “only” 2.53 times that of Earth. What that tells you is that Jupiter is much less dense than Earth, which makes sense given that it’s mostly hydrogen and helium. Its mass is neutered, so to speak, by its large radius.
So it’s all about the radius. That’s why HD 40307g doesn’t have a surface gravity eight times that of earth, but only 1.42 times.
Oddly enough all of the seven known potentially habitable exoplanets have nearly the same surface gravity, if the estimates are correct. Take a look:
Exoplanet, Mass, Radius, Surface Gravity
Gliese 581g 2.60 1.4 1.33
Gliese 581d 6.90 2.2 1.43
Gliese 667Cc 4.90 1.9 1.36
Kepler 22b 6.40 2.1 1.45
HD40307g 8.20 2.4 1.42
HD85512b 4.00 1.7 1.38
Gliese 163c 8.00 2.4 1.39
Fictional Planet 8.00 2.83 1.00
(All figures are expressed as multiples of Earth units. For example, Gliese 581g has 2.6 times Earth’s mass and 1.4 times its radius, for a gravity 1.33 times as much as ours. Source: Habitable Exoplanet Encyclopedia. Most of the masses and radii are estimates.)
You can see that except for Fictional Planet, which I’ll get to in a moment, they all have pretty much the same surface gravity despite huge differences in mass. Gliese 581g, with a mass 2.6 times that of Earth’s, has a surface gravity essentially the same as Gliese 667Cc’s, which has 4.9 times the mass of Earth.
It seems very curious to me that they all have such similar surface gravities. I have no idea why. Perhaps it’s an artifact of what our instruments can detect with current technology.
It’s cheering that these planets aren’t absolutely crushing, but still, they wouldn’t be easy for Earth visitors. On Earth I weigh 122 pounds (yes, I’m small.) On HD 40307g, I’d weigh 174 pounds. That would be just terrible for my back.
But it’s amazingly easy to imagine a super-Earth with a comfortable gravity. If a planet had eight Earth masses and 2.83 times the radius, its surface gravity would be exactly 1g. This is the “Fictional Planet” at the bottom of the table. Fictional Planet would be huge by Earth standards, with a circumference of 70,400 miles and an area eight times larger.
Does that mean we could land and take off with exactly the same technology we use here, assuming the atmosphere is similar? Actually, no. Another blogger, who who goes by the moniker SpaceColonizer, pointed out that Fictional Planet has a higher escape velocity than Earth. Put simply, escape velocity is how fast you have to go away from a planet to ensure that gravity can never bring you back. For Earth, escape velocity is about 25,000 miles per hour. Fictional Planet has an escape velocity 68% higher. That’s 42,000 miles per hour.
SpaceColonizer added a column to my table to account for escape velocity. Here’s the new table. SG = Surface Gravity, EV = Escape Velocity.
Mass Radius SG EV
Mercury 0.055 0.383 0.380 0.381
Venus 0.815 0.950 0.900 0.928
Earth 1.000 1.000 1.000 1.000
Moon 0.012 0.273 0.165 0.213
Mars 0.107 0.533 0.380 0.450
Gliese 581g: 2.600 1.400 1.330 1.360
Gliese 581d: 6.900 2.200 1.430 1.770
Gliese 667Cc: 4.900 1.900 1.360 1.610
Kepler 22b: 6.400 2.100 1.450 1.740
HD40307g: 8.200 2.400 1.420 1.850
HD85512b: 4.000 1.700 1.380 1.530
Gliese 163c: 8.000 2.400 1.390 1.830
Fictional Planet 8.000 2.830 1.000 1.680
If you feel no heavier on Fictional Planet, why do you have to go faster to get away from it? It’s a function of both mass and size. If Fictional Planet was four times Earth’s mass and had two Earth radii, its surface gravity would still be 1g, but you’d need to go only 41% faster to get away from it permanently. If you made it larger, four Earth radii, its escape velocity would be the same as Earth’s. (And its surface gravity would be ¼ of a gee.)
So surface gravity and escape velocity are both related to size and mass, but differently. Fictional Planet would be just as comfortable as Earth in terms of gravity, but more expensive to leave.
Basic rocket equations show that it would take four times as much fuel to get a given mass away from Fictional Planet. An emerging civilization on Fictional Planet would need four Saturn Vs to launch an Apollo-style mission to a moon. Say one for the command module, one for the LEM, one for the transfer engine, one for supplies. Or maybe just one big-ass rocket.
That is expensive – but we’re forgetting something about Fictional Planet. It has more resources. A single continent could have the entire surface area of Earth. An emerging civilization would have more land, more metals, more fossil fuels, and more room to sustain a large population. Think of it this way: if Fictional Continent was a unified nation, it could have seven billion people, which means a lot of taxpayers. You could easily have a Fictional NASA with four times the Apollo 11 budget. Relative to such a civilization, an Apollo 11-style mission could actually be easier than it was for 1960s America.
And it may be even easier than the equations make it sound. A spacecraft has go 25,000 mph to get to the Moon – it has to reach escape velocity to do it – but it has to go only 17,500 mph to get into low earth orbit. That’s 30% less.
On super-Earths, the difference between orbital velocity and escape velocity might not be 30%. It would depend on the height and density of the atmosphere and the planet’s rotational velocity, which gives you some some speed for free. But let’s just say, for the sake of discussion, that orbit is also 30% easier on Fictional Planet. Which means you only need 2.8 times as much fuel to get into orbit, not 4 times as much.
Once you are in orbit, it’s relatively easy to get the additional velocity you need to escape. No air resistance. You’re in less of a hurry, because you won't fall back to the planet anytime soon. The engineering is easier, because you don't need the same engine temperatures and pressures. In fact, you could achieve escape velocity with a slow but efficient ion engine. Deep Space 1 used a chemical rocket to get into orbit, and an ion engine to reach escape velocity so it could visit a comet and asteroid.
In short: once you can get off the planet, you have the technology you need to get away from the planet. It's just a matter of assembling the resources. And Fictional Planet has a lot of resources.
Far from being gravitational traps, super-Earths should be positive incubators of spacefaring civilizations. If I'm right about that, it would sharpen the Fermi paradox. If the galaxy has plenty of super-Earths in habitable zones, then you really have to wonder where everyone is.
Perhaps there’s geological reasons why such planets are unlikely. Maybe 1-gravity super-Earths would have too few heavy metals to sustain a civilization. Or maybe they wouldn’t have the plate tectonics to stabilize an atmosphere and biosphere for long enough for life to arise.
But still, I feel cheered. We already found seven super-Earths in habitable zones. It may not be long before we find one that has a gravity like ours and a tolerable orbital velocity. And it might be a super super-Earth.
Many thanks to @apollo18, who helped me with the math for this posting.
UPDATE: BoingBoing linked to this post and discussed it, here.