I have redone my original post on this, since I discovered that the
equations were completely garbled. I have now supplied the equations
as in-lined gif-files. If there is any problems with that, please tell me.
I have supplied answers to your questions/comments, Allen, in red.
[Re-post from 23.12.2007]
I simplify a bit by describing The Earth's orbit around the Sun as a circle
- not an ellipse. That is an approximation, but a good one for this purpose
- it will have no effect on the argument at hand.
The position vector of the centre of the Earth is then given by:
where we have put then Sun at (0,0,0), and the ecliptic plane (plane of the
orbit) is the x-/y-plane and the z-axis is perpendicular to that plane and
goes through the centre of the Sun. a is 1 astronomical unit (AU), the
average radius of the Earth's orbit around the Sun.
The first equality can also be written rc=(xc,yc,zc) , but the column
vector notation is more compact.
t is time, measured in the same unit as year (e.g., seconds, hours or days)
and the factor of 2*pi is there because the sine and the cosine go through
one full oscillation when the argument changes by 2*pi - Equation (1.1)
therefore goes through one cycle (orbit) in one year.
Equation (1.1), above, describes the translational orbit of the Earth around
the Sun. It describes how the position of the centre of the Earth changes
with time. Only position! - it says nothing about orientation.
The position vector of a particular point on the surface of Earth, at
the equator, is given by:
where the 23.5° is the tilt of the equatorial plane compared to the ecliptic.
The consequence of that tilt is that the point on the equator will go above
and then below the ecliptic during one day, as seen in the z-component of
the second term. The day is here the sidereal day.
The orbital rotaion does not take place in sidereal time.
That doesn't matter. The time variable, t, is continous - it can take on
any value, including fractions of a second. The period of the first term,
the orbit, is the year = 3.1557e7 s, while the period of the second term,
the rotation, is the sidereal day, which is 86164 s.
The orientation of the Earth is described by the vector going from the
centre of the Earth, to our chosen point on the equator, i.e., (1.2) - (1.1)
[equation (1.2) minus equation (1.1)]:
This describes the orientation of Earth with respect to the stars. In sidereal time
It changes in a cyclic manner, repeating itself every sidereal day.
There is no reference to anything happening on a yearly cycle.
Yes, in sidereal day/ time....which is not under consideration.
If you take a strobe light at a a rotating blade the period of the strobe will determine whether or not you see the blade to rotate clockwise or counterclockwise.
See my comment above, about time being continuous - no strobe-lights involved.
It should be obvious, that both the rotation described by Eq. (3) and
the translational orbital motion described by (1) are continuous motions
that happen all the time - e.g., the Earth does not jump forward every
time a sidereal day has been completed, instead the Earth moves ahead in
its orbit all the while it also spins on it's celestial axis.
As long as we are looking at sidereal time the two sets of vectors are balenced/ equivocated wrt nightly and annualy motions.
I don't know what you mean by this.
However, they are not in solar time. These equations describe the translational motions of the earth in orbit in sidreal time.
They describe the various parts of the Earth's motion at any time.
The translational motion of the Earth's orbit is described by Eq. (1.1).
but it only describes a fixed point on the earth’s orientation in sidereal time.
That Eq. (1.2) describes a fixed point on the earth’s surface was my intention - not
In sidereal time it does not jump forward but in solar time it would.
Time is continuous - no jumping anywhere.
Using a point at the center of the earth only makes my case. A fixed point on the earth progressively and radial moves around that point at the center while that point progressively and radial moves around the sun.
Okay - a line connecting the Sun and the centre of the Earth does indeed rotate.
The direction of the line changes with a period of one year, and that rotation is
obviously in the ecliptic plane.
That is all described by Eq. (1.1)
But that does NOT have ANY bearing on the orientation of the Earth.
The orientation of the Earth is measured with respect to the (far away) stars.
The orientation of the Earth changes with a period of 1 sidereal day and the
rotation is around the equatorial axis, as described by Eq. (1.3).
It is a compound motion not a simple single plane single axis rotation .
And what I have described with Eqs. (1.1-1.3), is exactly what you ask for.
So now you actually understand composite motions - very good.
The period in question is solar or 24 hour intervals,
The period of the Earth's rotation is the sidereal day. We are interested in the absolute
orientation in space, i.e., with respect to the (far away) stars.
these equations cannot show the orbital rotation
...because there is none - The orbit is a translation.
Eq. (1.3) shows that the only thing that changes the orientation of the Earth, is the
because there is none in the sidereal "strobe"/ interval. This would be true regaudless of distance a few feet or "infinity". The rotation is and can only be found in the solar "strobe" / interval/ 24 hours.....
Eq. (1.3) shows what goes on at all times - a rotation in the equatorial plane.
You can choose your "strobe" with any interval - till it looks like it is rotating
backwards, if you will, but it doesn't change the underlying motion.
The annual cycle is going to look the same as the nightly as long as you "strobe" the sidereal period. ( nightly).
That is because the only rotation is the daily rotation. Your annual effect arise from
seeing the sidereal daily rotation with a tropical daily strobe-light.
When you "strobe" a different interval the other and different effect will be seen if the cause of the effect realy exist.
If the effect exist then the cause exist. However, no effect then no cause.
Sounds okay to me.
I am going to have to settle this with somthing more then just logic and mathematics. Formal photo graphic experimental proof of what would/ should be seen is needed . This is what I will attempt to get accomplished.
It would be great if you could formulate your
position for all of us, so we can see
That's all for now.