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Re: Bulges





On Sat, 1 Mar 1997, Richard Grandy wrote:

The *bulges* do not depend on rotation, but the *tides* do. If the earth
and sun were fixed in space there would be bulges but not tides. If the
earth revolved around the sun but did not rotate, there would be two tides
a year.


I'm confused about terminology here. When I said that the bulges did not
depend on rotation I was referring to what are usually called the "tidal
bulges", and I was limiting my example to the earth-moon system (realizing
of course that there are smaller solar tidal bulges).

If, however, you limit the meaning of tides to the variation of water
level at a particular coastline location, then, with the stationary earth,
you'd not have those "tidal variations" of water level.

The reason the solar tides are smaller is worth mentioning, for students
often think that the size of the tide relates directly to the size of the
force from the external body. The gravitational force on the earth is
about 175 times larger from the sun than from the moon. But the tidal
deformation of the earth is not proportional to the gravitational force
but to the gradient of that force. Also, the ratio of the diameter of the
earth to the distance of the external body enters in.

I'm still looking for a *good* textbook discussion of this. Most books do a
lot of hand waving and use misleading language. One of the best
discussions I recall was in a Geology textbook. Here's some bad examples
of phrases to avoid when discussing the tides.

"Pulled towards..." Students interpret this as `moving towards'.

"Pull the water toward the moon." This implies motion. Actually,
the surface of the water on the near side of the moon doesn't move toward
or away from the moon, on average. Some authors may intend this to mean
that water is pulled away from low-tide regions into the high-tide
regions.

"Forget the rotational motion of moon and earth around each other and
consider them both in free fall toward each other." Students think of
relative motion toward each other (getting closer to each other). What the
author (Cliff Swartz, TIP) may mean is that they are falling relative to
the straight-line motion which they'd have in the absence of gravity.

"The effect of the moon on the earth is to pull the surface water toward
the moon on the near side and to pull the earth
away from the surface
water on the far side." (Swartz, TIP)

"The gravitational attractions of the moon and sun accelerate the whole
solid earth. These forces also accelerate the fluid water at the earth's
surface." (PP, 1981, p. 232) To speak of acceleration conjures up
images of increasing velocity in the student mind.

TIP means "Teaching Introductory Physics" by Cliff Swartz.
PP is "Project Physics".

The above examples, are, in my view, bad examples--how *not* to explain
the tides.

On the net, I found Phil Plait's "Home away from Home Page", which has a
good document about tides, as well as lots of other good astronomy
materials.

<LI><a href="http://www.astro.virginia.edu/~pcp2g/home.html";>
Phil Plait's Home away from Home Page</a>
<LI><a href="http://www.astro.virginia.edu/~pcp2g/tides.html";>
Tides, the Earth, the Moon.</a>
<LI><a href="http://www.astro.virginia.edu/~pcp2g/bad/bad.html";>
Bad Astronomy</a>

I have included this document by Phil Plait after my .sig. Notice that he
doesn't have to talk about center of mass, or centripetal effects to
explain the origin of the earth's tidal bulges. Of course the bulges are
modified by a number of effects, centripetal effects, being one. The
_ocean_ tides are modified by ocean depth and proximity to shorelines, as
well as ocean currents. To answer the question of why there are two tidal
bulges, on opposite sides of the earth, one doesn't need to complicate the
matter by getting into the details of _ocean_ tides. It's sufficient to
consider a water-free planet, and look at the deformation of it due to the
differential forces acting upon it.

Many students, when quizzed, show that they are totally confused by this
matter, and confuse the tidal bulges with the equatorial bulge (not
thinking in three dimensions). They wrongly think that there would be no
tidal bulges if the earth were not rotating. And I think that textbooks
are at fault for fostering such misconceptions.

-- Donald

......................................................................
Dr. Donald E. Simanek Office: 717-893-2079
Prof. of Physics Internet: dsimanek@eagle.lhup.edu
Lock Haven University, Lock Haven, PA. 17745 CIS: 73147,2166
Home page: http://www.lhup.edu/~dsimanek FAX: 717-893-2047
......................................................................


Every few months, one of a series of questions comes up on the USENET
group sci.astro involving tides, or the rotation of the Moon, or the
recession of the Moon from the Earth. In an effort to make a generic
answer to this, I have compiled a description of the tidal evolution
of the Earth Moon system. This description is long, and I am not
taking as much time as I should to edit it, but maybe after a while
I'll take another look at it and streamline it. For now though, I'll
let it be.

This missive explains the following:

* Why the Moon always shows the same face to the Earth.
* Why the Moon's rotation period is the same as the length of time
it takes to orbit the Earth (same as number 1, but phrased
differently).
* Why there are two tides a day.
* Why the Earth's rotation rate is slowing.

picture

The strength of gravity depends on the distance from the source. The
closer you are, the stronger the "pull" you feel. The Moon's gravity
acts on the Earth; but the diameter of the Earth is large enough in
relation to the distance of the Moon that the side of the Earth nearer
the Moon feels the Moon's gravity significantly more strongly than the
side of the Earth away from the Moon. If you could stand at the center
of the Earth you would feel the Moon's gravity somewhere between the
two.

This part is tricky, and is the hardest part of this explanation to
understand. A drawing of these forces looks like this:

--> ----> ------->
far center near
side of Earth side

where the arrows represent the force (and direction) of the Moon's
gravity on these three points of the Earth. Now, we measure the
gravity of the Earth relative to the center of the Earth; everywhere
on the Earth, the center is "down". In a sense, we see the center of
the Earth as "at rest". It is mathematically correct to then subtract
the force of the Moon on the center of the Earth from the force felt
on the near and far sides. This is called vector addition. If we do
that, our diagram will look like this:


far center near
side of Earth side

(Note that this drawing is not meant to be exact, but just to give a
feel for what's happening).

Now we see that in this sense, the Earth is stretched by the
difference in the Moon's gravity across the Earth. We call this effect
"tides". Tides are a differential force, that is, they result by the
difference in the force of gravity between two points.

That is why there are two tidal bulges on the Earth, one on the near
side, and one on the far side. Since water is more flexible than rock,
we see the tidal effect strongly in the oceans of the Earth, but
barely at all in the ground. However, the rock does bend, by as much
as 30 centimeters (about a foot) up and down twice a day!

As it it turns out, the tidal bulges do not line up exactly between
the center of the Earth and the Moon. Since the Earth rotates, the
bulges are swept forward a bit along the Earth. So picture this: the
bulge nearest the Moon is actually a bit ahead of the Earth-Moon line.
That bulge has mass; not a lot, but some. Since it has mass, it has
gravity, and that pulls on the Moon. It pulls the Moon forward in its
orbit a bit. This gives the Moon more orbital energy. An orbit with
higher energy has a larger radius, and so as the bulge pulls the Moon
forward, the Moon gets farther away from the Earth. This has been
measured and is something like a few centimeters a year.

Of course, the Moon is pulling on the bulge as well. Since the Moon is
"behind" the bulge (relative to the rotation of the Earth), it is
pulling the bulge backwards, slowing it down. Because of friction with
the rest of the Earth, this slowing of the bulge is actually slowing
the rotation of the Earth! This is making the day get longer. The
effect is small, but measurable. This is also why every few years
people that measure such things (chronologists?) need to add a leap
second to the year. There are more seconds per day since the Earth is
slowing down! (Note that the second is a unit based on how many times
a certain type of atom vibrates, and is not defined as a fraction of a
day.)

Eventually, the Earth's rotation will slow down so much that the bulge
will line up exactly with the center of the Earth and the Moon. When
this happens, the Moon will no longer be pulling the bulge back, and
the Earth's spin will stop slowing. But when this happens, the time it
takes for the Earth to rotate once will be slowed to exactly the same
time it takes for the Moon to go around the Earth once! If you were to
stand on the Moon and look at the Earth, you would always see the same
face of the Earth.

Does this sound familiar?

It should. Since Earth's gravity is much stronger than the Moon's, the
tides from the Earth on the Moon are much stronger than the Moon's
tides on the Earth. The Moon has tidal bulges just like the Earth, and
so it too was slowed by the Earth's pull on its nearer bulge.
Eventually, the Moon's rotation was locked so that it took the same
time to spin once on its axis as it takes to go around the Earth. This
is why we always see the same face of the Moon! And this happened to
the Moon before the Earth because the Earth's tides are so much
stronger.

If this is hard to picture, grab two oranges; one for the Earth and
one for the Moon. Let one go around the other, first without any
rotation and then letting the "Moon" rotate just once on its axis for
every time it goes around the "Earth". See how if the Moon does NOT
rotate, then eventually we see all sides? Therefore the Moon does
rotate, but it does very slowly: once a month!

Once the Earth is rotationally locked with the Moon, there will be no
more evolution of the system from either the Earth or the Moon.
However, there are tides from the Sun, which are actually about half
as strong as those of the Moon (the Sun is much farther away than the
Moon, but a LOT bigger). This will continue to affect the system, but
at this point you're on your own!

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