Will There Ever Be An Earth With Rings?

Jupiter may be the god but Saturn is the Adonis of our Solar System. The majestic rings of Saturn, have for centuries been the subject of spiritual and scientific debate. But how did Saturn get them in the first place?Why do planets like Saturn, Uranus and Neptune have rings while our Earth does not? In this article thou shalt find thy answers.

History:

Galileo Galilei was the first to observe the rings of Saturn in AD 1610. However, because of the crudeness of his telescope he incorrectly deduced that the rings were actually two moons on either side.

Galileo’s drawing of Saturn, 1610 (above) and 1616 (below)

In   AD   1659,  the  Dutch  astronomer  Christiaan  Huygens  used  an improved  telescope  and  solved  the  mystery  of  Saturn’s  “arms”  .  After careful  observations  he  correctly  deduced  that  the “arms” were actually a ring system.

The genius, James Clerk Maxwell, in AD 1859, wrote an essay on the structure of Saturn’s rings and after 60 long pages of rigorous mathematics showed that they were not solid structures but were made of disconnected, independent   particles.   His   predictions   were   finally   confirmed   by   the Voyager Missions(1977) and the Cassini-Huygens Missions(1997).

Voyager 2’s image of Saturn’s rings
Radio occultation images of Saturn’s rings from the Cassini Huygens Space Missions

The Cassini Huygens Space Missions not only settled the debate for the structure of Saturn’s rings but also provided us with useful information on their origin. There is considerable evidence in support of the theory that the rings formed when one of Saturn’s moons came within its Roche Limit. The moon got shredded by the tidal forces of Saturn.

The remainder of this article is dedicated towards an elementary discussion on tidal forces, a derivation of the expression for Roche Limit and the fates of our Celestial Couple – Earth and Moon.

Tidal Forces:

According  to  Newton’s  Law  of  Gravitation,  the  gravitational  force between two point masses is attractive, and spherically symmetric (depends only on the distance between the two bodies). In fact it goes as 1/r2 , r being the distance of separation. On account of this, points that are closer to the source are attracted more strongly than those that are farther apart. In other words, a gradient exists in the force field and this results in what are known as tidal forces. Tidal forces result in bodies getting stretched and are responsible for various phenomena like tides, tidal locking, ring formation, etc. For intense gravitational force fields, those that exist near Black Holes and Neutron Stars, tidal forces result in spaghettification: the gradient of the force field is so high that even over extremely small distances, the variation in the strength of the forces is immense and the body gets elongated like a spaghetti and disintegrates into oblivion.

Comet Shoemaker-Levy 9 breaking up under Jupiter’s tidal forces

Tidal forces due to the moon result in high and low tides on earth.

Moon’s gravity differential field on the earth’s surface. The bulges signify high tides, the dips signify low tides

Roche Limit:

In the previous section it was mentioned that tidal forces are responsible  for  ring  formation.  However,  this happens only if the body is closer  to  the  source  than  its  Roche  Limit.  The  Roche  Limit  is  therefore defined  as  the  distance  within  which  a  celestial  body held together by its own force of gravity, will disintegrate due to a second celestial body’s tidal forces  exceeding  the  first  body’s  gravitational  self  attraction.  The  term  is named after Edouard Roche, who first calculated it in 1848.

Images showing the fate of a body under the influence of tidal forces. Notice how the body gets shredded into a ring within the Roche Limit (white curve). Order: Left to right; Top to bottom.

The following subsection gives a derivation of Roche Limit.

Derivation:

Derivation of the Roche Limit Expression

To avoid complications, we deal with completely rigid spherical bodies. We shall ignore any non-uniformities in shape, size, composition and mass distribution of the bodies. The yellow one will be designated as the primary. The tidal forces are due to the gravitational field of the primary and their effects are felt by the secondary, which is assumed to be in free fall.

Say we have a mass u on the surface of the secondary. This mass experiences two forces: the tidal force, FT , due to the primary and the gravitational pull of the secondary, FG .

The expression of FG is obtained from Newton’s Law and it is given as,

It is always directed toward the center of the secondary.

The expression for FT is obtained by finding the difference in the primary’s pull on the center of the satellite and on the edge closest to the primary,

Applying the approximations, r R and R < d , we get,

and it is directed towards the primary.

The Roche Limit is reached when FT = FG and this gives,

Substituting,

and,

the expression becomes,

More accurate expressions exist, and can be derived, by considering fluidity, inhomogeneity, distortions due to tidal effects, etc.

In the not so distant past, many held the view that the rings of Saturn were as old as the planet itself, and had formed from the planet’s protoplanetary accretion disc. Substantial evidence was provided in support of the contrary by the Cassini Huygens Space Missions. It is now widely accepted that all of Saturn’s rings with the exception of the E-Ring and Phoebe Ring, have formed in this way. Infact, the rings of Uranus and Neptune also fall within their Roche Limits.

The  Roche  Limits  for the Earth-Moon system, rigid and fluid, come out  to  be  9,492  km  and  18,381  km  respectively.  The  actual  value  lies somewhere in between. Therefore, if the moon ever happens to be within its Roche Limit, the Earth’s tidal forces will rip it into pieces and the rotational kinetic energy of the moon will cause it to form a ring around the earth. In the event of such an occurrence, the sky will look somewhat like the picture below,

If the Earth had a Ring too

Would have been a sight to behold. Too bad, the chances of occurrence of such an event are pretty slim. By the way, don’t know about you, but I like our moon right where it is.

By Soumyajit Chatterjee, Department of Physics, IISER Bhopal.

References:

  1. Classical Mechanics, John R. Taylor
  2. h ttps://en.wikipedia.org/wiki/Roche_limit
  3. h ttps://en.wikipedia.org/wiki/Tidal_force

Image Sources:

  1. https://www.cnet.com/pictures/earth-if-you-like-it-put-a-ring-on-it-pictures/
  2. h ttps://attic.gsfc.nasa.gov/huygensgcms/Shistory.htm
  3. h ttp://pds.jpl.nasa.gov/planets/captions/saturn/saturn.htm
  4. http://photojournal.jpl.nasa.gov/catalog/PIA07874
  5. http://hubblesite.org/newscenter/newsdesk/archive/releases/1994/21/image/b
  6. Krishnavedala
  7. https://en.wikipedia.org/wiki/Roche_limit
  8. https://en.wikipedia.org/wiki/Roche_limit#/media/File:Roche_limit_(with_small_mass_u).svg
  9. h  ttps://www.universetoday.com/126698/what-would-earth-look-like-with-rings/

About the author:  Soumyajit is presently a 4th-year physics major at IISER Bhopal. His areas of interest include Non-Linear Dynamics, Quantum Information Theory, Statistical Mechanics, Machine Learning and Relativistic Dynamics. Besides being a lover of physics, he is passionately zealous about the armed forces of India and wishes to be a part of it someday. He likes to sketch, dance and code and plays football, badminton and carrom. Soumyajit is a great fan of Batman, John Constantine and Phir Hera Pheri.
Another blog by the author that you might want to read: https://theqriusrhino.wordpress.com/2020/07/26/oscillating-chemical-reactions/

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