In Which the Blogger Overanalyzes a Children’s Book and Loses All His Readers

Si tu aimes un fleur qui se trouve dans une étoile, c’est doux, la nuit, de regarder le ciel. Toutes les étoiles sont fleuries. (If you love a flower found on a star, it is sweet at night to look at the heavens. All the stars are blooming.)
–The Little Prince

In my Ph.D. defense presentation, I quoted from just one book: The Little Prince (Le Petit Prince) by Antoine de Saint-Exupéry. The quotation I used wasn’t picked for scientific content—the story isn’t known for scientific accuracy, after all—but I was thinking about the setting when I wrote Monday’s post about Dactyl, an asteroid moonlet only a 1.6 kilometers (about 1 mile) across. After all, the Little Prince lives on an asteroid the size of a house, known as B-612. The asteroid is home to three volcanoes and a flower, which (incidentally to today’s post) the Prince falls in love with. He later visits other asteroids, which are drawn as similar in size.

The book isn’t science fiction: it’s more a melancholy semi-philosophical fantasy, ostensibly for children, though I imagine adults love and appreciate it even more. In other words, it’s unfair to do what I’m about to do: look at what it would take for an asteroid the size of a house to have Earthlike gravity, and what life would be like on such a tiny world. However, today is my birthday, so I am indulging myself with a frivolous post—albeit one that gets into the subject of gravity, and tides, and the nature of astronomical bodies.

Gravity, Mass, and Size

Gravity in Isaac Newton’s theory is a force acting between two masses. (General relativity, the more modern theory of gravity that uses the geometry of spacetime instead of a force extending through space, gives the same results when gravity is weak. More on this later.) The farther apart the two masses, the weaker the force: doubling the distance results in a force 1/4 as strong. However, there’s a quirk—gravity acts as though each mass were concentrated at a point, so when you stand on the surface of Earth, the “separation” is the radius of the planet. This means if you are at the top of Everest, the force of gravity is slightly weaker than if you are standing in Death Valley. Similarly, since Earth isn’t perfectly spherical, gravity is slightly stronger at the poles than it is at the equator. (To make things even more complicated, Earth’s rotation also influences the force we feel!) However, we’ll assume Earth and the asteroids are spherical, just for simplicity.

While the force of gravity depends on your mass, the action of gravity—known as the strength of the gravitational field—is the same on every object. If you neglect air resistance, every object falls with the same acceleration: 9.8 meters per second faster for each second it falls. (Meaning: if it starts from rest, after one second, it will be moving at a speed of 9.8 meters per second; after two seconds, it will be falling 2 × 9.8 = 19.6 meters per second; and so forth. If you prefer American units (ptui!), use 32 feet per second per second for the acceleration instead.) The gravitational field depends on the mass of the planet or other object and where you are measuring acceleration relative to the object’s center.

Because gravity gets weaker with larger distances, if we want an asteroid the size of a house with Earthlike gravity, it will need to be very massive: far more massive than a typical asteroid can be. Though we don’t have a measured mass for Dactyl, if it’s a fairly typical composition and density (say 2.5 times the density of water, which is slightly less than half Earth’s density), its mass will be around 5×10¹² kilograms (roughly 5 million tons). While it may sound like it’s extremely massive, that’s about 9×10^-13 times Earth’s mass—that’s 0.0000000000009 times. Assuming this mass value is approximately correct, Dactyl’s gravitational field strength is about 0.009% of Earth’s, so if you weigh 100 pounds on Earth, you’d weigh just 0.009 pounds on Dactyl.

Life on Asteroid B-612

So could a world like Asteroid B-612 exist? To see that, let’s change the rules a bit: now we want to fix the gravitational field strength while shrinking the radius down to Little-Prince size. The figure at the right shows these relationships: an object 1/10 the size of Earth must have a mass 1/100 of Earth’s to have the same gravitational attraction at its surface. That’s a steep decrease: by the time we’ve reached Dactyl’s size (about 10^-4 Earth’s radius), the the mass required is 10^-8 Earth’s, or 100 millionth. While that’s pretty small compared to Earth, it’s also a lot more massive than Dactyl is in real life!

Obviously we don’t have an accurate size for B-612, but judging from the book cover above, it’s probably about 2 meters in diameter (assuming the Prince is 1 meter tall). That puts B-612 at about 2×10^-7 times Earth’s radius, meaning its mass is about 10^-14 Earth’s. While it’s less massive than the real Dactyl, it’s still very massive for its size.

The implications are obvious: to make an asteroid with Earthlike gravity, you’d need to build it of rock far denser than anything found on Earth. Dactyl would need a density 10,000 times Earth’s density, and B-612 would need to be about 10 million times the density of Earth. That’s a lot more dense than the Sun, and even more dense than a white dwarf, which is the compact core of a star similar to the Sun that has exhausted its usable supply of hydrogen and helium. In other words, there is no ordinary material that has the density needed to make B-612 possess Earthlike gravity. (The situation is worse with neutron stars, also the cores of stars, but far more massive—neutron stars are much more dense even than our hypothetical asteroid.)

As an aside, even as crazy as B-612 is, its gravity is not strong enough that we need general relativity. We can show this by calculating the hypothetical asteroid’s Schwarzschild radius—the distance at which nothing can escape, not even light. If the Schwarzschild radius is bigger than the radius of the asteroid, then B-612 is actually a black hole. The formula for the Schwarzschild radius is remarkably simple:meaning that a object with the same mass as our Sun has a Schwarzschild radius of 3 kilometers, something twice as massive will be 6 kilometers, and something half as massive will be 1.5 kilometers. Since B-612 has a tiny fraction of Earth’s mass, which is in turn a tiny fraction of the Sun’s mass, the Schwarzschild radius of B-612 is really minuscule: much smaller than the nucleus of an atom.

Fit to Be Tide

One interesting final point about B-612 again relates to its tiny size. On Earth, we humans are but ants crawling across the vast globe: even the tallest person is short compared to the distance between the surface and center. That means the gravitational field is about the same on our heads as on our feet. The Moon, on the other hand, experiences a slightly stronger force on its near side than its far side, which over a period of billions of years has slowed its rotation until it now presents one face to Earth. This effect is known as tidal locking. The Moon also famously exerts a similar tidal force on Earth, which raises tides twice per day in the oceans.

The Little Prince, standing about a meter tall on an asteroid itself only a meter in radius, will feel Earthlike gravity only on his feet. His head, situated twice as far from the center of the asteroid, will experience 1/4 the gravitational acceleration. So, simply making B-612 impossibly dense doesn’t sort out the problems of making it livable. (I haven’t even gotten the issue of keeping a breathable atmosphere!)

Postscript

All fantasy and much of science fiction depart from the physics of the real world, so my analysis isn’t criticism. The environment we live in on our Earth is a property of many things—a combination of mass, size, composition, location in the Solar System, and the like. Though the story of gravity is only part of the tale, it’s something interesting to look at as we explore asteroids and other objects, looking for the environment of life. There may not be a Little Prince with his flower living on an asteroid, but understanding why not is a part of science just as much as observing Enceladus.

So what did I quote from The Little Prince in my Ph.D. defense? I was speaking about cosmology, and specifically the dark matter and dark energy in the universe, so I selected a short sentence first spoken by the Fox and later repeated by the Prince himself.

L’essentiel est invisible pour les yeux. (The essential is invisible to the eyes.)