Nick and Geoff explain why there are two tidal bulges on the Earth – one below the Moon and one on the other side

Slide1

Unfortunate Astronaut Fred falling head-first into a black hole. The red arrows indicate the gravitational force acting different parts of Fred’s body. Drawing Nick Lomb

The cause of tides has been known since the days of Isaac Newton (1642-1727), but it is sufficiently complex to have become a matter for debate amongst the astronomers at Sydney Observatory in recent weeks. Here Nick and Geoff try to explain why the Earth has two tidal bulges due to the Moon – one below and one on the other side.

INTRODUCTION

Consider the sad fate of Astronaut Fred who is falling head-first into a black hole and towards oblivion. As indicated in the instantaneous picture above there is a larger force on his head than on his body and less on his feet. So we can say that his head is being pulled away from his body and his body from his legs.

Slide2

Astronaut Fred falling into the black hole, in a frame of reference that accelerates at the same rate as Fred’s centre of mass. Drawing Nick Lomb

We can look at Astronaut Fred in a more sophisticated way. If we choose a frame of reference that falls with Fred’s centre of mass (CoM). Then, as Fred’s fall is largely taken care of by the frame of reference, what we are left with on the above diagram are the remaining forces that are slightly different from the main gravitational force on Fred’s CoM. As his head is accelerating towards the black hole even in this frame of reference, it experiences a force away from his CoM. Similarly, Fred’s legs are accelerating away from the black hole in this frame of reference, they experience a force away from the CoM.

These forces that represent differences from the main force acting on a body’s CoM are called tidal forces.

Slide3

A simple instantaneous picture explaining tides, drawing Nick Lomb

Now let us consider tides on the Earth’s oceans caused by the Moon under the simple instantaneous picture. The forces on the Earth’s CoM must be balanced. Gravity is pulling towards the Moon and so we have to introduce the fictitious centrifugal force (connected with rotation) away from the Moon.

On the side of the Earth directly below the Moon gravitational pull from the Moon is stronger as the distance is shorter while centrifugal force is less. This unbalance leads to bulge towards the Moon. On the other side of the Earth gravitational pull towards the Moon is less as the distance is greater while centrifugal force is more. Again there is a bulge, this time away from the Moon.

Slide4

A more sophisticated picture of tides on Earth in a frame of reference co-moving with the Earth’s centre of mass. Drawing Nick Lomb

As with Astronaut Fred we can look at tides in a more sophisticated way if we choose a frame of reference that co-moves with the Earth’s CoM. Again that accounts for the main acceleration towards the Moon and what we are left with in the diagram above are the tidal forces, that represent slight differences to the main force on the CoM.

The oceans below the Moon are accelerating less towards the Moon than they would need to according to the greater gravitational pull they experience than the CoM. Hence there is a tidal force towards the Moon and a tidal bulge. Similarly on the other side of the Earth acceleration is too much according to the gravitational pull at that location. Hence there is a tidal force away from the Moon and a tidal bulge.

DETAILED EXPLANATION

Although rather simplistic, a force is simply a push or a pull and throughout the Universe there are only four types(1). From the rotation of galaxies to the inside of the atom, four simple types of force reign supreme. They are in order of decreasing strength:

1. The Strong Nuclear Force
2. The Weak Nuclear Force
3. The Electromagnetic Force, and
4. The Force of Gravity.

Amazingly, the Strong Nuclear Force is 10 to the power of 40 times stronger than gravity yet its range is stunningly small and limited to the realm of the nucleus. Gravity however, a mere wimp in comparison acts over unlimited distances with ever diminishing strength as we shall see.

Image1

Isaac Newton

Along with Einstein, one of the greatest contribution to the study of gravity occurred in the period of 1665 to 1667 when plague forced Sir Isaac Newton (1642-1727) to flee London. Most of us have heard the story that he was struck upon the head by a falling apple and wondered what made it fall. It is very likely this is apocryphal, but perhaps he did see an apple fall whilst in a contemplative mood. By applying his second law of motion which states “The acceleration of an object is proportional to the net force applied to it and inversely proportional to its mass” Newton realised that some force must pull or push the apple down. Might this same force act on objects a long way away like the Moon?

He eventually devised the theory of Universal Gravitation in which gravity is the force of attraction between all objects of mass to every other object of mass. His formula for Universal Gravitation is as follows:

Fgravity = G*m* M/r2

In it, the force of gravity between two objects of mass (m) and (M) is equal to their product multiplied by the constant of Universal Gravitation (G) divided by the distance between them squared. If the mass of the objects does not change then we can see that the force between them is inversely proportional to the square of their distance from one another. Using this formula alone, the Sun exerts more than 170 times the force of gravity on the Earth than does the Moon, after all the Earth orbits the Sun. So why is the tidal influence of the Moon so much more noticeable than that of the Sun? Consideration of the above formula reveals that objects other than point sources such as voluminous planets and stars will experience a difference in gravity potential from one side to the other. Algebraically we can derive that tidal forces, a difference in force across an object due to gravity, follows an inverse cubed relationship. Clearly distance becomes much more critical in this case.

Ftidal = 2*G*M*m*R/r3

Where (M) and (m) are the mass of the primary object and its satellite with a radius (R) at distance from one another of (r) and assumes that R/r is very small. The difference in gravity potential or the gravitational field across the object is the key to tides and not simply the magnitude of the force. As a result the influence of the Sun on tides drops to 46% of the effect of the nearby Moon.(2)

Image2

Johannes Kepler

The tidal pull of the moon raises and lowers the rocky surface of the Earth by about 0.3m (3) every day while the more elastic water can be raised by up 15m depending on surrounding terrain but in which direction?

At this point we need to turn to Kepler’s Third Law.

P2/a3 = k

Where (P) is the period, (a) is the semi major axis and (k) is a constant. Therefore for any orbiting object the period of its orbit squared multiplied by the semi major axis of its orbit cubed equals a constant.

Geoff 1

Kepler’s Third Law in Action, drawn by Geoff Wyatt

In the above diagram Geoff is in a stable orbit of (O) to the left. (B) is the centre of the planet and describes the orbit. As the planet is a solid object (A) according to Kepler’s third law is orbiting too quickly for its orbital radius (O to B- [B to A]) and tends to push inwards creating a bulge. (C) is orbiting too slowly for its orbit (O to B + [B to C]) and the resulting force tends to push or bulge planet outwards.

A more detailed analysis utilizing Newton’s gravitation which can be derived from Kepler’s Laws also explains two tidal bulges as the result of differences in forces caused by a distant object.

Tides 1

Vector Analysis of Tidal Forces 1, from Donald Simanek

In this image taken from Donald Simanek’s page on “Misconceptions about Tides” vector analysis for four points A,B, H and D reveals the following.(4)

Object (G) exerts a gravitational force on (M) that follows the inverse cube law as stated above. All vectors are directed towards (G).
Consider a person standing at (H). There will be a greater force on their head than their toes and the tidal force (T), will be given by the difference, TH = FH1 – FH2. The net force is shown by the vector subtraction on the lower left with the resultant force towards (G).
Now consider a person on the other side of (M) at (A). The force on their toes is greater than on their head and the difference in forces is TA = FA1 – FA2 but as can be seen by the vector diagram on the upper left the resultant force is away from (G).

For a person at (D) the forces are almost identical and parallel as a result the tidal force points towards the centre of (M) and at (B) the resultant vector is tangential to (M).

Image 3

Vector Analysis of Tidal Forces 2, from Donald Simanek

If this process were conducted for many points on the surface the above diagram would result. Two tidal bulges are clearly seen on (M), one on the side toward the attracting mass and the second on the far side of (M).

Image 5

Mont St Michel in France

A wonderful example of tides and their uses can be seen at Mont St Michel in France where the incoming tide completely surrounds the island even protecting its inhabitants from invaders in the past.

Curiously some scientists believe we owe our very existence to the tides. For without them the seas would not rise and fall several times a day creating an opportunity for sea creatures to gain foothold on land in the tidal zone between high and low tide.(5)

Apart from the rising and falling of the each objects surface the other major tidal force effect is that the Moon was gravitationally locked to the Earth when in a molten state and remains so locked. Being tidally locked the Moon has reached its lowest energy state and rotates once on its axis every rotation around the Earth.

Image 4

Tidal recession of the Moon

As both objects bulge towards the other but not in a perfectly straight line as a result of the Earths rapid rotation, a torque or turning force due to tidal friction arises that slows the Earth’s rotation and pushes the Moon away from the Earth in order to conserve angular momentum. The diagram above shows that as the Earth’s tidal bulge leads the Moon by about 6 hours the forces between the Moon and the bulge balance as per Newton’s Third Law. However the overall effect of the two forces is to slingshot the Moon ahead hence pushing it out while at the same time opposes the Earth’s rotation and slows it down. Yes the days are getting longer!

Image 7

A crater chain on the Moon

The Moon is also, clearly, a long suffering victim of tidal forces from other places in the Solar System. In the image below of Davy Crater, a crater chain has most likely been caused by the tidal disruption of a loosely bound comet, which then collided with the Moon. Pre impact, the comet passed by a more massive object, quite probably Jupiter, and was subjected to tidal forces which pulled it apart. The loosely bound objects then peppered the Moon in quick succession.

Image 8

Comet Shoemaker-Levy 9 collisions with Jupiter

A more recent and spectacular example of a similar event was the tidal disruption of Comet Shoemaker Levy No.9 in 1992, before colliding with Jupiter in 1994. As the comet approached Jupiter, the leading side was subjected to a stronger force than the trailing side. As it was comprised of loosely bound particles, it simply fell apart under the tidal difference and formed a train of 21 large pieces. Two years later, after it passed the Sun, its altered orbit put it on a collision course with Jupiter and led to a spectacular series of impacts.

Tidal forces, by the very nature of gravity, are seen throughout the Solar System. The second largest object in the Solar System, Jupiter, has at least 63 moons (6). Its inner most moon, Io, is subject to the most violent volcanic activity witnessed because of the changing gravitation fields or tidal forces. A complex gravitational dance between Jupiter and the other large nearby moons subject Io to constantly changing forces. As Io flexes between 50 and 100m over short periods enough friction is generated to heat the interior to the point of volcanism.

Another large moon, Europa, is also tidally heated (7) to a point where, just a few metres below its icy surface, alkaline liquid water is known to exist which would otherwise be frozen. Earth does not, therefore, have exclusivity on liquid water.

Even large scale features such as galaxies can be affected by tides. If they get too close to one another they can be subject to disruptive tidal forces with perhaps the most beautiful example being the Antennae galaxy.

The Antennae

The Antennae, galaxies in collision, Hubble Space Telescope image

In this Hubble Space Telescope Image, the preceding and following arms are the result of tidal forces between the two galaxies. As they approach one another stars are being drawn ahead into a leading arm, while others across the galaxy, where the tidal force is less, are left behind at the same time. Eventually, tidal forces will drive them to be roughly spherical in nature, a shape that would have pleased Aristotle immensely.(13)

Image 9

Aristotle and Plato

Geoff Wyatt and Nick Lomb, Sydney Observatory

BIBLIOGRAPHY

(1) http://cnx.org/content/m14044/latest/
(2) http://en.wikipedia.org/wiki/Tide
(3) http://www.astronomycast.com/solar-system/episode-47-tidal-forces/
(4) http://www.lhup.edu/~DSIMANEK/scenario/tides.htm
(5) http://www.math.nus.edu.sg/aslaksen/gem-projects/hm/0102-1- phase/TIDESONLIFE.htm
(6) http://www.nineplanets.org/jupiter.html
(7) http://www.resa.net/nasa/europa_life.htm
(8) http://physics.fortlewis.edu/Astronomy/astronomy%20today/CHAISSON/GLOSSARY/GLOSS_R.HTM
(9) http://www.britannica.com/eb/article-9063949/Roche-limit
(10) http://en.wikipedia.org/wiki/Roche_limit
(11) http://en.wikipedia.org/wiki/Roche_limit
(12) http://www.astronomycast.com/solar-system/episode-47-tidal-forces/
(13) http://www.astronomycast.com/solar-system/episode-48-tidal-forces/

IMAGE CREDITS

Sir Isaac Newton http://www.virtualmuseum.ca/Exhibitions/Annodomini/THEME_15/IMAGES/J991825.jpg
Johannes Kepler http://www.nasa.gov/vision/universe/starsgalaxies/kepler_prt.htm
Vector Analysis of Tidal Forces 1 http://www.lhup.edu/~DSIMANEK/scenario/tides.htm
Vector Analysis of Tidal Forces 2 http://www.lhup.edu/~DSIMANEK/scenario/tides.htm
Mont St Michel http://www.gites-de-bretagne.com/images/montstmichel_lge.jpg
Tidal Recession of the Moon http://odin.physastro.mnsu.edu/~eskridge/astr102/spinslow.gif
Davy Crater Chain http://www.lpi.usra.edu/publications/slidesets/3dsolarsystem/slide_17.html
Shoemaker Levy 9 and Jupiter http://www2.jpl.nasa.gov/sl9/
Antennae Galaxy http://hubblesite.org/newscent er/archive/releases/1997/34/image/a/format/large_web/
Plato and Aristotle http://en.wikipedia.org/wiki/Image:Sanzio_01_Plato_Aristotle.jpg

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