Finding minimum rotation period

On 17 Jun 2006 18:17:43 GMT, Ricky Romaya
wrote:

Hi,

If given a planet's mass, radius, and density, how can I find out the
minimum rotation period, or maximum rotation speed, assuming the planet is
rotating around it's own axis and is a perfect sphere?

Wikipedia only give a hint that for celestial bodies, the body's tensile
strength is the one supplying the centripetal force, and if the angular
speed is much greater in respect to it's density, the body will break. I
couldn't find any equations which relates the tensile strength or density
with centripetal force.

Please help

TIA
Here are some interesting facts, using the earth as an example..
The gravity inside the earth is proportional to the radius, beginning
with 0 at the center and increasing linearly to the surface. So
g(r) = g0*r/R
It's proportional to r.
Similarly the centrifugal acceleration is proportional to r:
gcent = W^2r
So equating the two
g0*r/R = W^2*r
W = sqrt(g0/R) rad/sec (1)
2pi/W = period for 1 revolution (84.4 min)
(1) produces a net zero gravity everywhere in the earth or planet.
That is the same period as a low flying shuttle orbit. At this angular
rate, there is no gravity anywhere (assuming constant density) in
earth so it would tend to come apart.

John Polasek

Ricky Romaya wrote:
"oriel36" wrote in
ups.com:

The actual mechanism for the deviation of the Earth's shape from a
perfect sphere is differential rotation in the flexible and molten
interior across the Entire length of the Earth's axis and certainly
not from the dead center as contemporary notions have it*.

For once,do something productive and pick up on latitude dependent
differential rotation and apply the lessons learned from observing
rotating celestial objects with exposed flexible conditions such as
the Sun's plasma and the correlation to a deviation from a perfect
sphere.It then becomes a matter of adjusting the Earth's interior
composition,its viscosity,rotation rate from Equator to pole ect. -

http://www.astronomynotes.com/starsun/sun-rotation.gif

Just think 'across the entire length of the Earth's axis' and you will
get to love the mechanism for the planetary deviation from a perfect
sphere.

Could you point me to some inet resources where I can learn more about
this?

The resourses are scattered all over the internet but not in an
organised way and certainly not linking the dynamics of the Earth's
shape and crustal motion under a common mechanism - differential
rotation perpendicular to the planetary rotational axis.

There are informal discussions on Saturn's large deviation from a
perfect sphere but none exist for the Earth either through
unfamiliarity or oversight.It becomes easier to assign a differential
rotation to the Earth by considering the molten and flexible interior
alone and then grafting in the fractured surface crust later as the
rotational dynamics create a basis for the motion of the crust.Even in
outlines,these dynamics look far more exciting that remaining with
stationary Earth/convection cell mechanisms.

I suppose you are best ploughing your own course on this exciting
material,think of it as updating the already known correlation between
rotational dynamics and the Earth's shape with data based on a
flexible and molten interior and replacing vague descriptions based on
models which reference the bulge from the dead center of the Earth
rather than the more accurate conception based on rotational dynamics
perpendicular across the entire length of the Earth's axis.

Good luck in your endeavors.

In article ,
Ricky Romaya writes:
If given a planet's mass, radius, and density, how can I find out the
minimum rotation period, or maximum rotation speed, assuming the planet is
rotating around it's own axis and is a perfect sphere?

Wikipedia only give a hint that for celestial bodies, the body's tensile
strength is the one supplying the centripetal force...

If Wikipedia really says this about _planets_, then it's wrong. As
someone else noted, for large masses, tensile strength is negligible
compared to gravity. Also, a planet near breakup will (in general)
be very far from spherical.

The general solution of this problem is complicated. As others have
written, the maximum rotation speed is where the gravitational force
at the surface equals the centripetal force needed for a circular
orbit, but calculating the gravitational force for a non-spherical
body is not always trivial. You need to know the shape of your
planet and how density changes within it, and those depend on what
the planet is made of (in particular on the "equation of state" of
the material, i.e., how its density changes as a function of
pressure). Composition may not be uniform, so you may have multiple
materials to consider. I'd be surprised if there is an analytic
solution in the general case.

If you are willing to accept simplifying assumptions, there can be
analytic solutions. For example, the case of uniform density results
in the "Maclaurin spheroids,"
http://www.phys.lsu.edu/astro/H_Book...aclaurin.shtml
There are also "Jacoby ellipsoids," but I forget exactly what
simplifying assumption is made to derive those. (Anybody remember?)

By the way, a few people who have responded to this thread don't seem
to know much about physics. Be careful about which answers you take
seriously.

--
Steve Willner Phone 617-495-7123
Cambridge, MA 02138 USA
(Please email your reply if you want to be sure I see it; include a
valid Reply-To address to receive an acknowledgement. Commercial
email may be sent to your ISP.)

On 19 Jun 2006 12:56:49 -0400, (Steve Willner)
wrote:

In article ,
Ricky Romaya writes:
If given a planet's mass, radius, and density, how can I find out the
minimum rotation period, or maximum rotation speed, assuming the planet is
rotating around it's own axis and is a perfect sphere?

Wikipedia only give a hint that for celestial bodies, the body's tensile
strength is the one supplying the centripetal force...

If Wikipedia really says this about _planets_, then it's wrong. As
someone else noted, for large masses, tensile strength is negligible
compared to gravity. Also, a planet near breakup will (in general)
be very far from spherical.

The general solution of this problem is complicated. As others have
written, the maximum rotation speed is where the gravitational force
at the surface equals the centripetal force needed for a circular
orbit, but calculating the gravitational force for a non-spherical
body is not always trivial. You need to know the shape of your
planet and how density changes within it, and those depend on what
the planet is made of (in particular on the "equation of state" of
the material, i.e., how its density changes as a function of
pressure). Composition may not be uniform, so you may have multiple
materials to consider. I'd be surprised if there is an analytic
solution in the general case.

I don't see that it is complicated at all.

It is plain that the dirt would peel off at the equatorial bulge,
where only the surface condition governs.

In Eq. (1) g0 = W^2R so you measure g0 at the equator (regardless
of shape) and knowing Rbulge you can calculate W^2. They can't be
looking for more than a simplest criterion, which would seem to be
loss of gravity's coercive powers. Eq. (1) appears a sufficient
determinant regardless of shape.

If you are willing to accept simplifying assumptions, there can be
analytic solutions. For example, the case of uniform density results
in the "Maclaurin spheroids,"
http://www.phys.lsu.edu/astro/H_Book...aclaurin.shtml
There are also "Jacoby ellipsoids," but I forget exactly what
simplifying assumption is made to derive those. (Anybody remember?)

By the way, a few people who have responded to this thread don't seem
to know much about physics. Be careful about which answers you take
seriously.

John Polasek
http://www.dualspace.net

In article ,
John C. Polasek writes:
I don't see that it is complicated at all.

It is plain that the dirt would peel off at the equatorial bulge,
where only the surface condition governs.

In Eq. (1) g0 = W^2R so you measure g0 at the equator (regardless
of shape) and knowing Rbulge you can calculate W^2.

How do you know Rbulge? I suggest you try writing down the equation
for density as a function of radius and latitude, given some assumed
equation of state. (If you do this right, you will find yourself
writing down the equation of hydrostatic equilibrium in cylindrical
coordinates, then solving it for some equation of state.) Once you
have that, write an equation for gravitational force. Remember that
the mass distribution is not spherically symmetric nor is density
constant.

Perhaps there's some simplification I don't see, but the general case
looks hard to me. Of course it's easy to get an approximate answer
by assuming spherical symmetry. That may not be a bad approximation
if much of the mass is concentrated near the center of the planet.

--
Steve Willner Phone 617-495-7123
Cambridge, MA 02138 USA
(Please email your reply if you want to be sure I see it; include a
valid Reply-To address to receive an acknowledgement. Commercial
email may be sent to your ISP.)

On 20 Jun 2006 12:29:36 -0400, (Steve Willner)
wrote:

In article ,
John C. Polasek writes:
I don't see that it is complicated at all.

It is plain that the dirt would peel off at the equatorial bulge,
where only the surface condition governs.

In Eq. (1) g0 = W^2R so you measure g0 at the equator (regardless
of shape) and knowing Rbulge you can calculate W^2.

How do you know Rbulge? I suggest you try writing down the equation
for density as a function of radius and latitude, given some assumed
equation of state. (If you do this right, you will find yourself
writing down the equation of hydrostatic equilibrium in cylindrical
coordinates, then solving it for some equation of state.) Once you
have that, write an equation for gravitational force. Remember that
the mass distribution is not spherically symmetric nor is density
constant.

Perhaps there's some simplification I don't see, but the general case
looks hard to me. Of course it's easy to get an approximate answer
by assuming spherical symmetry. That may not be a bad approximation
if much of the mass is concentrated near the center of the planet.

I think you brought up the bulge complication. To quote the OP:

"If given a planet's mass, radius, and density, how can I find out the
minimum rotation period, or maximum rotation speed, assuming the
planet is rotating around it's own axis and is a perfect sphere?"

It's a school boy question and needs a schoolboy answer.