Assuming an absolute reference point in the universe what can be said
of the absolute velocity of a point on earth taking into account
geographical position, altitude, earth's rotation, orbital speed,
motion of the solar system, motion of the galaxy and all other speeds
and "directions" involved. Has there been any mathematical formulation
of how this vector changes in time? That is, what are the varying
accelerations that a terrestial point might be subjected to?

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Request for Question Clarification by thx1138-ga on 13 Jun 2002 20:03 PDT

Hi arminius and thanks for this great question, the answer to which
could be amazingly complicated!

There is not and cannot be a fixed point from which to view the Earth
unless you specify where that point is in reference to something i.e.
from a particular crater on the moon.  Then using that particular
location on the moon you can comment about your observations of a
point on Earth or any other astronomical body. You could also say
that, from a particular spot on Mars a particular point on Earth
appears to move in a certain way.
Let me know where you would like the hypothetical reference point to
be and I’ll give it some serious thought and research.


Thanks very much

THX1138
“ Einstein taught us there's no fixed point in the universe; if you
say object A is moving with respect to stationary object B, I can say
B is moving with respect to A”
http://newark.rutgers.edu/~jlynch/Papers/pyrrho.html

“I realized that I was in the predicament of Archimedes, who vainly
sought for a fixed point in the universe”
http://www.lucidcafe.com/library/96jul/teslaauto03.html

Hello arminius, and thanks for your interesting question!

As others have pointed out in the comments, there is no distinguished
"absolute reference point" in the universe. No doubt you understand
this, because you have asked us to "assume" an absolute reference
point.

For the purposes of this question, I will use the cosmic microwave
background radiation to supply a frame of reference, and will
summarize the components of the earth's velocity relative to that
frame of reference.

The velocity of a point on earth's equator due to the earth's rotation
is proportional to the distance of that point from the centre of the
earth. The equatorial radius of the earth is 6378 km. To that radius
we must add the altitude above sea level of the point in which we are
interested. For example, for a 1000 meter mountain we would add 1km to
the radius. We then multiply by 2 pi to get the rotational
circumference, and divide by the number of seconds in the sidereal day
to get the velocity of that point due to the earth's rotation.

The siderial day, the time taken for the earth to present the same
point towards the "fixed" stars, is 86164.10 seconds. Therefore, the
velocity at the top of a 1000 meter mountain due to the earth's
rotation is approximately 465 meters per second or 1675 km/hr.

For a point which is not on the equator, multiply by the cosine of its
latitude. For example, London is at latitude 51 degrees North, so the
velocity of London is 465 times 0.629, or 293 meters per second.

This is only an approximation, because the earth is not a perfect
sphere. If you wish to correct for this, details can be found at Eric
Weisstein's "World of Astronomy" on the "Earth" page:
http://scienceworld.wolfram.com/astronomy/Earth.html

There are many other corrections that can be applied to take into
account peturbations of the earth's rotation, including the Chandler
Wobble
http://scienceworld.wolfram.com/astronomy/ChandlerWobble.html
the Milankovitch Cycles
http://scienceworld.wolfram.com/astronomy/MilankovitchCycles.html
and a slipping about its rotation axis
http://scienceworld.wolfram.com/astronomy/VariationofLatitude.html
however these effects are all small.

There is a tidal effect due to the moon and, to a lesser extent, the
sun. In addition to the ocean tides, there is a tidal effect on solid
land of about 0.2 meters approximately twice every 25 hours:
http://www.henry-cort.hants.sch.uk/find/science/space/

The earth's orbital speed around the sun is about 29786 meters per
second (based on an average orbital distance of 149.6 million
kilometers and a year of 31.56 million seconds):
http://scienceworld.wolfram.com/astronomy/Year.html

The earth's orbit around the sun is elliptical, and corrections can be
made for this:
http://www.analemma.com/Pages/EllipticalOrbit/EllipticalOrbitMath/EllipOrbitMath.html

The gravitational effect of the moon and of the sun's planets also
affects the earth's orbit.

The axis of the earth's rotation is not the same as the axis of its
revolution around the sun. The difference between these (of 23.5
degrees) causes the earth's seasons.

The solar system is part of our galaxy, and is about 30,000 lightyears
away from its center. It travels around that center about once every
200 million years, at an average speed of about 230 km per second:
http://www.sunspot.noao.edu/PR/answerbook/motion.html

Our galaxy is part of a group of galaxies known as the Local Group,
which are close enough together to have a significant gravitational
effect on each other. Our galaxy is moving towards the center of the
Local Group at around 40 km per second:
http://www.sunspot.noao.edu/PR/answerbook/motion.html

Now let's consider the cosmic microwave background radiation, which is
thought to be remnant radiation from the early stages in the evolution
of the universe. This radiation comes from all directions, but the
strength varies according to direction in a pattern that suggests we
are moving relative to the matter which last scattered the background
radiation.

Our local group of galaxies is moving through the cosmic microwave
background radiation at about 600 km per second:
http://www.nobel.se/physics/laureates/1978/wilson-lecture.pdf
(page 18)

However, our sun is moving through the cosmic microwave background
radiation at only 370 km per second, due to its relative movement
within the local group.

It is not possible to produce a mathematical formulation to show how
the earth's overall velocity changes with time. One reason is that the
earth's rotation and orbit are slowing down and will change
significantly within the time of one galactic revolution. Another
reason is that the various gravitational bodies interact, and this
interaction will change as their positions change. As dannidin-ga
pointed out in the comments, it is not even possible to mathematically
formulate the motion of THREE gravitationally-interacting bodies.

Nevertheless, you can get a feel for the overall motion by
superimposing the orbital rotation of the earth around the sun onto a
370 km per second linear motion of the sun relative to the background
radiation. The other velocities are either much smaller or are
changing very slowly.

I hope you find this information useful.


Additional link:

Astronomical constants from Eric Weisstein's "World Of Astronomy":
http://scienceworld.wolfram.com/astronomy/Earth.html


Google search strategy:

"astronomical constants"
://www.google.com/search?hl=en&lr=&q=%22astronomical+constants%22

london latitude
://www.google.com/search?hl=en&ie=UTF8&oe=UTF8&q=london+latitude

tides "solid land"
://www.google.com/search?q=tides+%22solid+land%22&hl=en&lr=&start=20&sa=N

universe "frame of reference"
://www.google.com/search?hl=en&lr=&q=universe+%22frame+of+reference%22

"cosmic microwave background radiation"
://www.google.com/search?hl=en&lr=&q=%22cosmic+microwave+background+radiation%22&btnG=Google+Search

"hubble constant"
://www.google.com/search?hl=en&lr=&q=%22hubble+constant%22

"group of galaxies" "particular velocity"
://www.google.com/search?hl=en&lr=&q=%22group+of+galaxies%22+%22particular+velocity%22


Regards,
eiffel-ga

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Clarification of Answer by eiffel-ga on 15 Jun 2002 02:03 PDT

Hi arminius,

Thanks for explaining your motivation for asking this question.

In your comments you mention that you were in an aircraft and wondered
"what manner (speed and direction) was I personally whizzing around
and how did that change with time?"

If the aircraft is moving at around 0.25 kilometers per second, this
is less than a thousandth of the 370 kilometers per second that the
solar system is moving relative to the cosmic microwave background
radiation.

Yet when it comes to change of direction, the aircraft can make a 360
degree turn in a minute or so, which is more than a thousand times
faster than the earth is rotating.

So the dominant factors are the astronomical speeds and the local
changes of direction.

Regards,
Roger

Hello eiffel-ga

Perhaps an explanation of how the question arose in the mind of a
mathematically naive biologist.  I was flying from Boston to San
Francisco walking from forward to aft in a United Airlines jet when
the question occurred to me: how could I describe my motion with
reference to the cosmos? Starting with the pace I was walking,
subtracted from the speed of the jet, adding the rotational speed of
the earth at my present latitude, adding the orbital speed of the
earth around the sun, taking into account the speed and direction of
the solar system within our galaxy, the motion of said galaxy with
respect to some fixed point in the universe, etc. In other words, in
what manner (speed and direction) was I personally whizzing around and
how did that change with time? What, in effect, are the accelerations
that I was and am undergoing ?While I can more or less understand the
concept of speed, the component of the vector, direction, eludes me. 
I cannot visualize how "direction" would be described in a cosmic
sense, since the cardinal points of the earthly compass cannot come to
the rescue. So while the first few steps might be manageable, not
knowing the magnitude of the movements of galaxies boggles me.

Thanks for the rapid, indeed instantaneous response.  I appreciate the
thoroughness of your answer and your clarity of exposition. Before you
proposed the microwave background as a point of reference I was going
to suggest the quasar called BR 1202-0725.

Best regards
arminius-ga

I think you could take the center of mass of the universe as your
fixed point and the rest frame of the center of mass of the universe
as your inertial frame.  I don't know how far Earth is from that point
or how fast it is moving relative to that point.

I don't think you could take the centre of mass of the universe as
your fixed point. The thing is that to the best of our current
cosmological knowledge, the universe is curved. Compare it to the
surface of a sphere, which is 2-dimensional and curved in the 3rd
dimension. Now try finding the centre of the surface of a sphere.
Within its own two dimensions. You cannot. Now, you may object, that
doesn't mean that there isn't a centre *of mass* for that surface.
However, if the mass on that surface is relatively (i.e. relative to
the order of the size of the sphere) homogeneously spread out, you
won't be able to find one. Now I guess that the universe's mass on
such a large scale comes close to being homogeneously spread out, but
I should ask a 'real' cosmologist about that some time or look it up.
There are, of course, lumps, which we call galaxies, and lumps of
lumps, yes indeed, but then, I never heard of these lumps of lumps
lumping together in a specific "corner" of the universe.

I'll take up the question where thx1138 left it, that is, if we are
given some point in the universe as a reference point, how to
calculate the location of some given point on Earth as a function of
time. This can be done, IN THEORY, that is, the answer is the solution
of some differential equation describing the dynamics of motion of all
the bodies of the universe under their mutual interacting forces. In
Newtonian mechanics (which we know to be false, of course) one would
simply write down Newton's law F=mA for each of the bodies in the
universe, where F is the force, in this case the sum of the
gravitational pulls of all the other bodies, m is the mass of the
body, and a is the acceleration (the second derivative of the
location). Once we solve this system of equations, we need to
transform our system of coordinates to a system where our reference
point becomes the fixed point 0.
Like I said, Newtonian mechanics is unsuitable for treating problems
at this scale, so really we would have to use the equations of
Einstein's general relativity theory, which are much more complicated,
but can equally well be written down by a competent professional (i.e.
not me).
So why did I mention Newtonian mechanics? To demonstrate that even in
this much simplified case (not to mention that we neglected electrical
forces and all sorts of other weird phenomena such as stars exploding
etc.), the answer can only be computed IN THEORY, whereas in PRACTICE
this becomes, well, practically impossible. Even for a problem
concerning the motion of THREE celestial bodies, there is no
analytical expression (that is, an explicit solution of the equations
of motion given by a formula) for the motion of the bodies - this is
the famous three-body problem (whose amusing history is described in
detail in the book The Problems of Mathematics by Ian Stewart).

Hope this helps,
dannidin

p.s. some potentially useful search terms:
Newton's laws of motion
general relativity (theory)
the Michelson-Morley experiment (a physics experiment designed to
measure the speed of the Earth relative to the hypothetical absolute
fixed point of the universe, whose famous failure led eventually to
the understanding by Einstein that there can be no such thing and to
his discovery of relativity theory - see for instance
http://www.drphysics.com/syllabus/M_M/M_M.html
http://www.aip.org/history/einstein/emc1.htm)

Einsteins relativity does not provide a frame of reference in space
(everything is relative), but recent
empirical discoveries of the cosmic background radiation from the COBE
satellite have established
one. This microwave radiation is spread almost uniformly through space
and is the cooled remnants
of the heat from the ancient big bang. By measuring the temperature in
different directions and
finding a doppler effect creating red and blue shifts in the heat, it
has been established that our
 galaxy is moving through space at about 1 million miles an hour (fast
!) in the direction of space
 that we see occupied by the constellation of Leo. 

We also know that the sun is orbiting with our rotating galaxy - about
once every 200 million years.
The sun also has its own localised movement relative to the nearby
stars - a kind of bouncing up
and down through the plane of the galaxy - about once every 10,000
years, which may be related
to the frequency of ice ages.

Movements of the earth within the solar system are well-known.

I don't know the equations, but take all these movements into account
and you will get your
answer !

There is an interesting point nobody addressed yet.

Arminius originally suggested using the quasar BR 1202-0725 as a point
of reference. Doing so would give a somewhat surprising result:

Taking this quasar as a reference point, the earth would be moving at
282,000 kilometers per second!

This enormous speed comes from the fact that our universe is
expanding. As the universe expands, the galaxies move away from one
another. And the further an object is, the faster it is receeding from
us.

As Quasar BR 1202-0725 is one of the most distant objects known, it
moves away from us at the incredible speed of 282,000 kilometers per
second (94% the speed of light). And if we take the quasar as a point
of reference, the earth will be the one moving at this mind-boggling
speed.