###### By Joshua Winn, Princeton University

## The International Space Station is orbiting only 400 kilometers above the Earth’s surface. That’s only 1.06 Earth radii, well within the Roche limit. So, should the astronauts be concerned? Maybe you’re even getting really worried: “I’m standing right on the Earth’s surface, at one Earth radius! I’m going to get torn apart!” But the fact is there is no need to panic.

### Chemical Bonds vs. Tidal Force

The Roche limit was derived by setting the tidal force, which tries to pull the body apart, equal to the attractive gravitational force trying to hold it together. But there are other ways for a body to hold together, besides gravity. There are also chemical, or material forces, that give rocks their rigidity.

The silicon atoms in a rock are not held in place by gravity. They’re stuck together with chemical bonds, which are ultimately electromagnetic forces at the atomic level. Comet Shoemaker-Levy 9 broke apart because it was a loose conglomeration of rocks and chunks of ice, with hardly any material strength. So, there wasn’t much holding the chunks together besides gravity.

### It’s All about Gravity

The lesson is that the Roche limit is only relevant for objects that are held together mainly by gravity. And, we can’t take it too seriously: that’s for the ideal case of a frictionless fluid. Material forces allow a body to come closer than that official limit.

We can find little moons, smaller than about 500 kilometers, nestled right within the rings, inside the Roche limit. Why is that? It must be because those objects are held together mainly by material forces, and not gravity.

In fact, you can tell if an astronomical object is gravitationally bound just by looking at it. If gravity is the dominant force, it will be a sphere. It’s because each piece is attracted to every other piece. Left to its own devices, gravity draws everything inward toward the center of mass, smoothing out any lumps making a perfect sphere.

In contrast, chemical and material forces are very local; they act only between neighboring molecules, or surfaces in direct contact. So, a body held together by those forces can be any shape—an egg, a potato, a person—depending on the history of how the pieces came together. But if we make the object bigger and bigger, gravity becomes more important and eventually dominates over the chemical and material forces.

This article comes directly from content in the video seriesIntroduction to Astrophysics. Watch it now, on Wondrium.

### The Need for Massiveness

Let’s do an order of magnitude calculation to see how big a body has to be, for gravity to mold the shape into a sphere. Suppose we have a rock with a characteristic size of *R* and mass *M*. Now, let’s make it slightly larger by adding a single silicon atom. Which is more important: the molecular forces that bind the silicon to the minerals on the surface, or the gravitational attraction of the atom to the entire mass of rock?

It’ll be easiest to compare the relevant amounts of energy. We know that the energy levels of electrons in atoms are always on the order of a few electron volts. That’s the energy scale of atomic bonds. The energy scale for material forces tends to be an order of magnitude lower, as rocks aren’t perfect crystals; they’re ragged collections of crystals, and the interactions between them are weaker.

So, let’s say the energy released is, on average, a 0.1 of an eV per silicon atom. Meanwhile, the gravitational potential energy that’s released when we add mass of *m* is of order *G ***×** *M ***×** *m* over *R*. For gravity to be more important, we need that to be much larger than a 10th of an electron volt.

### Finding the Radius

That gives us a condition on the rock’s mass over radius, *M *over* R*. We’re wondering about the critical size, so let’s write *M* as volume times density, and then solve for *R*. What we find is that *R* must exceed the square root 3 over 4*pi* times 0.1 eV over *Gm-rho*.

Time to insert some numbers. The density of rock is around 3 grams per cubic centimeter or 3000 kilograms per cubic meter, and the mass of a silicon atom is about 28 proton masses or 4.7 **×** 10^{-26} kilograms. Plugging those in, along with all the constants, leads to a critical radius of about 600 kilometers.

Now, we can’t take that number 600 too seriously, as this was only an order-of-magnitude estimate. But based on this calculation, we would expect objects much larger than that to be sculpted into spheres, by gravity, while much smaller objects can have irregular shapes.

### How Stars Can Be Torn Apart

Stars, themselves, can get torn apart, if they violate the Roche limit. Sometimes a star will be orbiting another star at what seems like a safe distance, for billions of years, but then the star expands, it becomes a giant star. This lowers the star’s density, which in turn causes the Roche limit to reach out to larger separations. We can see that from the equation: if the rho in the denominator goes down, the minimum distance goes up.

A star can also get torn apart if it gets too close to a black hole. We’ll have more to say about black holes later. For now, be aware that they really exist, and in particular, many galaxies, including our own, have an enormous black hole sitting at the center. If a star happens to venture too close to the black hole, well, that’s the end of the star.

### Common Questions about How Big a Body Has to Be to Form a Sphere

**Q: Why aren’t astronauts affected by the Roche Limit?**Because their body particles are held together by chemical bonds, which are much stronger than tidal forces or any other form of gravitational force, so the tidal forces can’t tear their body apart when they reach the Roche limit.

**Q: What happens to gravitational and material forces in astronomical objects when their mass increases?**If astronomical objects get more massive, gravity becomes more significant within them, so it will eventually dominate the chemical and material forces. Thus, tidal forces at the Roche limit can affect big astronomical objects.

**Q: How can stars get torn apart?**Stars can get torn apart if they violate the Roche limit, for example, when they get close enough to a massive black hole.