Quote of the Day
Whenever you find yourself on the side of the majority, it is time to pause and reflect.
— Mark Twain
Introduction
Figure 1: Image of Phobos, the
largest moon of Mars (Wikipedia).
While listening to the audio book The Search for Exoplanets: What Astronomers Know, I heard the lecturer (Professor Joshua Winn) mention the Roche limit and gave a simple approximate formula for evaluating it. The Roche limit provides a lower boundary on how close a satellite may revolve around a planet or star. It is based on the idea that the gravitational and centrifugal forces of the planet work to pull a satellite apart, while the self-gravity of the satellite tends to hold it together. The Roche limit is where these forces are in balance – any closer and the satellite's gravity will be weaker than the centrifugal force plus the planet or star's gravity. Within the Roche limit, the satellite is subject to forces that tend to break it apart. Satellites moving inside the Roche limit are thought to be one way that planetary rings are formed.
You occasionally hear the Roche limit discussed for satellites in our solar system, like Phobos of Mars (Figure 1), where the satellite is near the Roche limit and may begin the process of coming apart in as little as 10 million years. There are actually a number of satellites in our solar system that are under threat of being torn apart by the centrifugal and gravitational forces induced by their orbiting close to a more massive companion. At least one comet has been observed breaking up due to gravitational forces (Appendix B).
In this post, I will be reviewing the Wikipedia's article on the Roche limit, and I will derive an interesting approximation that Professor Winn mentioned in his audio lectures. I find the approximation interesting because it expresses the Roche limit in terms of orbital period rather than distance.
Background
Definitions
- Tidal Force
- In the context of the Roche limit, a tidal force arises because the gravitational force exerted by one body on another is not constant across it; the nearest side is attracted more strongly than the farthest side. The differential nature of the tidal force tends to distort the shape of the smaller body and in extreme cases can even cause the smaller body to break up. (Source).
- Hydrostatic Equilibrium
- There are three forces on a satellite that are balanced at the Roche Limit – self-gravity of the body, internal pressure, and gravitational tidal forces (Source –Chapter 14). The Wikipedia has a good discussion of the concept.
- Roche Limit
- Minimal orbital distance compatible with hydrostatic equilibrium within a planet (Source –Chapter 14).
Roche Limit Statement
You usually see the Roche limit expressed in one of two ways: (1) for a rigid satellite, (2) for a fluid satellite. Planetary astronomers focus on the fluid satellite formula, but the rigid satellite formula is much simpler to derive and it illustrates the concepts. The derivation for a fluid satellite is more complex because it assume that the satellite becomes distorted under tidal forces – two examples of distorted bodies from within our solar system are shown in Appendix A.
Rigid Satellite
The rigid-body Roche limit assumes a spherical satellite – the irregular shapes caused by the tidal deformation of a body are neglected. It is assumed to be in hydrostatic equilibrium. These assumptions, although unrealistic, greatly simplify calculations.
| Eq. 1 |  |
where
- ρM is the density of of the large body (planet or star).
- ρm is the density of of the satellite.
- RM is the radius of the planet or star that the satellite orbits.
- dRigid is the rigid satellite Roche limit – within this limit, a rigid satellite will experience tidal forces sufficient to tear it apart.
Note that the leading coefficient in the rigid body derivation is sometimes given as 1.26. A more accurate derivation (given in the Analysis portion) gives 1.44 for the leading coefficient.
Fluid Satellite
Equation 2 shows the Roche limit for a fluid body, i.e. a body with no internal binding forces. Note that the Wikipedia has a coefficient of 2.44, which was derived by Roche and is commonly seen. I am using the form presented on the McGill University web site, which has a coefficient of 2.423.
| Eq. 2 |  |
where
- dFluid is the fluid satellite Roche limit.
Analysis
Rigid Satellite Roche Limit Derivation
Figure 2 shows my derivation of Equation 1. The derivation of the Roche limit for rigid body is much simpler than for a fluid body because the fluid body will undergo major distortion into an ellipsoid shape, which is analytically much more difficult.
Figure 2: Derivation of Roche's Rigid Satellite Result.
Fluid Satellite Roche Limit Derivation
The derivation of Equation 2 is much more involved, and I will not go through the details here. The best derivation I found was on the McGill University web site. There is also a good reference on google books (Figure 3).
Application Example
Figure 4 shows an example I copied from the Wikipedia entry on the Roche limit. For my example, I have chosen to use the rigid satellite formula from the Wikipedia (my Equation 1) and the fluid formula from the McGill University web site (my Equation 2), mainly because I understand the derivations of both formulas. The normalize Roche limit is simply the Roche limit expressed in terms of the radius of the planet or star.
Figure 4: Worked Example.
Interesting Alternative Roche Limit for Fluid Satellites
Derivation
Professor Winn expressed the Roche limit in terms of a minimum orbital period (Equation 3). The derivation of this form of the Roche limit simply requires applying Kepler's third law.
| Eq. 3 |  |
Figure 5 shows the details of my derivation.
Figure 5: Derivation of the Equation 3 Approximation.
Worked Example
Mars' Phobos provides an interesting application example for Equation 3. Figure 6 shows my calculations using Equation 3 and we can see that Phobos nearing the fluid Roche limit.
Figure 6: Phobos Example Using Equation 3.
Since Phobos is in a decaying orbit, some astronomers speculate the Phobos may only have 10 million years of left as an intact body. Here is an interesting quote from this article on the "doomed" satellite Phobos.
But Phobos won't zip around the red planet forever. The doomed moon is spiraling inward at a rate of 1.8 centimeters (seven-tenths of an inch) per year, or 1.8 meters (about 6 feet) each century. Within 50 million years, the moon will either collide with its parent planet or be torn into rubble and scattered as a ring around Mars.
Conclusion
I have been looking for a good excuse to go through the derivation of the Roche limit and the effort was worth it. I was able to duplicate the results shown on the Wikipedia and on a university astronomy department web site. I was also able to derive a useful approximation that I heard in the audio book version of The Search for Exoplanets.
After writing this post, I found a good paper online co-written by Professor Winn on this topic.
Appendix A: Examples of Satellites Possibly Undergoing Tidal Stress.
Figure 7 shows two satellites that some planetary astronomers feel may be undergoing tidal stress.
Figure 7: Satellites Atlas (Jupiter) and Pan (Saturn) (Source).
The following quote from Quora does a nice job describing why satellites like these are hanging together.
Pan and Metis are held together by tensile forces. Tensile strength of a body is the maximum stress it can withstand before being pulled apart by stretching. Had their core been weaker, they would have disintegrated. The tidal forces affecting the bodies is why both the satellites are irregularly shaped. It is presumed that the primary's tidal forces can actually lift an object off the satellites' surface. Naturally, since the satellites are so close to their respective planets, there is massive tidal deceleration at play. This means that the satellites are gradually spiraling towards the primaries, owing to the decay of their orbits. The tidal forces are constantly tugging at the satellites. Pan's surface (as we've discovered from Cassini) consists of a large amount of porous material that it has accreted. The particles are weakly bound by their self-gravity, and had there been no other satellites, would eventually shear out forming large clumps before they disintegrate and the particles join other clumps.
Appendix B: Example of Comet Broken Up By Tidal Forces
Figure 8 shows a picture of comet Shoemaker-Levy 9, which was broken up by Jupiter's gravity in 1992.
Figure 8: Shoemaker-Levy Comet Broken Up By Jupiter's Gravity (Source).