Henri Wilson wrote:

Hold a circle (or an ellipse) in front of you at any angle.
Rotate your head until you find an axis in the plane of the
circle that is horizontal to the line between your eyes,
and is also perpendicular to the LOS. (one always exists)
ALL the radial velocities and the accelerations around the
orbit are then multiplied by the same factor, cos(pitch),
where the pitch angle refers to the rotation around the
above axis.

Rotating one's head is irrelevant. The rotation that you
describe (A "roll" of either the head or the projected
ellipse) simply puts the long axis of the projected ellipse
on the viewer's X axis. That is convienient but has no
effect on the process of multiplying radial velocities and
accelerations around the orbit by a factor of cos(pitch).

My apologies. I was wrong when I said 'The rotation that
you describe (A "roll" of either the head or the projected
ellipse) simply puts the long axis of the projected ellipse
on the viewer's X axis.' That is always true of circles,
but usually not true of ellipses. Actual ellipses generally
do not end up with the long axis of the projected ellipse
being horizontal following such a rotation.

The rotation still appears to be irrelevant, though.

You said this previously and I do not understand why George
did not point out its irrelevancy at that time.

Do I understand your terminology correctly as saying that
the "pitch" of an orbit is zero when seen edge-on and 90
degrees when seen face-on?

Yes...but the rotation is about an axis in the edge-on
position....that axis lying perpendicular to the LOS and in
the plane of the orbit.

If I may clarify your clarification: You are now talking
about a second rotation. The first, as you said, can be
accomplished by rotating one's head. It has the effect of
making the axis for the second rotation horizontal WRT the
viewer. The second rotation is the pitch of the orbit.

However, neither rotation is ever actually necessary or
carried out in your analysis. The first is never done.
The second, AFAICS, is not actually a rotation, but the
value that you use for the pitch of the orbit.

It is ALWAYS POSSIBLE TO FIND SUCH AN AXIS, no matter what
the orbit configuration wrt Earth.

Perhaps I misunderstood your reason for describing the
first rotation. It has no effect on the maths used, but
it *does* show which axis you are referring to. Is that
the reason for bringing it up? Simply to explain which
axis you mean? You can do that just by saying 'A line in
the plane of the orbit perpendicular to the line of sight'.
This omits the bit about rotation to horizontal position,
because you never execute such a rotation. You could say
that the line passes through the center of the ellipse,
but that isn't really needed either since any line in the
plane of the orbit and perpendicular to the line of sight
will serve. Using a centerline just looks nicer.

That is a very minor objection. I raise it only because
you described the rotation to horizontal as though it were
part of your analysis process, but it isn't.

If so, your term "pitch" means the same as "inclination",
which is the term everyone else uses in astronomy. Though
it is often measured as angular deviation from face-on
rather than from edge-on. That is how it is used in arXiv
astro-ph/0507420.pdf (Table 1, "Orbital inclination, i")

To double-check that we are talking about the same thing,
see the illustration of "yaw", "pitch", and "roll" near the
top of this page:

http://mtp.jpl.nasa.gov/notes/pointing/pointing.html

I have tried to explain before that I have redefined pitch
and yaw to make the programming of this stuff possible. My
method is 100% correct and effective. For the purpose of
brightness variation and measurement, one angle can be
eliminated by simply 'rotating the horizontal', ie.,
one's head.

The thing is, you are not starting with the ellipse at an
angle, so there is no angle to eliminate. If you started
with measurements of the actual orientation of the projected
ellipse in the sky, then it would be convenient to rotate
it so that a particular axis is horizontal WRT the observer.
But you are not starting with measurements of the actual
orientation of the projected ellipse, so there is nothing
to rotate. Instead, you are constructing an ellipse from
scratch, and not specifying orientation. (No reason to.)

Every orbit, eliptical or circular can be described in this
way. ...an edge on orbit multiplied by cos(pitch)...or
'inclination' as you call it.

Rotating one's head to change the orientation of an apparent
ellipse is trivial, and has no effect on the maths used.

If the actual shape is a circle, then the axis found is
the major axis of the projected ellipse. In that case,
of course, there is no need for a third rotation in yaw.

If the actual shape is an ellipse, then the axis found will
usually be at an angle to the major axis. In that case, it
appears that your resulting pitch angle will generally be
somewhat larger than the conventional inclination. I have
not anayzed this fully and am not certain of the result.

The conventional inclination is the angle between the line
of sight and the plane of the orbit, while your 'pitch' is
the angle that the orbit would need to rotate around a line
in the plane of the orbit and perpendicular to the line of
sight in order to become edge-on. I'm not certain, but it
appears that that angle would usually be larger than the
conventional inclination.

The only difference this makes that is obvious to me is the
one you are already aware of: It changes the value for yaw.
I have not attempted to analyze that at all.

To verify what I am saying, I suggest you make a paper
cutout of an ellipse, stick it at some odd angle onto the
end of a rod and hold it up in front of you. If you rotate
the rod (representing the LOS) you will see that at one
particular angle there will be an axis in the orbit plane
that lies perpendicular to the LOS and parallel to the line
between your eyes (the new horizontal).

In that position, the orbit can be rotated around THAT AXIS
through an angle (my 'pitch') into an edge on position. I
define YAW as the angle between the major axis of the ellipse
and the LOS when the orbit is in that edge on position. My
'zero yaw angle' is also defined differently ...for
programming reasons.

Thus, both acceleration and velocity can be simply multiplied
by cos(pitch) to reduce their component in the direction of
the observer. The effect is to simply reduce the height of my
predicted brightness curves but not their **shapes**, which
are determined solely by eccentricity and yaw angle.

Do you mean that the effect (a reduction in height of your
brightness curves) is a purely mathematical manipulation for
convenience in your program, or that you predict an actual
reduction in brightness? If the latter, what you are saying
is that the brightness is reduced from what is expected given
the inclination derived from observation.

In general, pitch is a factor in determining brightness.
A factor does not always reduce the resulting value. You
say that multiplying by cos(pitch) reduces the brightness
because you *want* and *need* the brightness to be reduced
in this particular case in order for your program to
produce results which match the observations.

Pitch is a factor in brightness, not a brightness-reduction
mechanism. It cannot be ignored or left out of the equation
without rendering the resulting value meaningless.

Note: It is not possible to resolve the pitch angle from
a point source of light and I know of no method that can
determine the pitch component involved in a measured
velocity. So my radial velocity figures automatically
represent (orbital velocity x cos(pitch).

Conventional analysis gets the inclination of J1909-3744
to better than two significant digits, via two separate
methods, which exactly agree with each other.

Leonard