Re: Pulsar Nav accuracy

Quoting hal@buffnet.net:

> Hello John L.,
>   Thanks for the explanation below, but of course, I've more questions
> to
> ask then...  ;)
> 
> >Hal,
> >	 A roundabout and overly simplistic answer is;
> >1 point establishes a sphere that you are on the 
> >surface of, I presume the radius can be determined.
> >2 points establish a line, and therefor you are on the
> >
> >'surface' of a circle defined by the intersection of 
> >the two angles to the two ends of the line.
> >3 points determine a plain, and only two locations
> >can satisify the requirements for the required angles
> >to the determined locations. (I.E. plus or minus
> >angles)
> >4 points are needed to determine the plain and
> >determine
> >if you are above or below the level of the plain.
> >5+ points are better.
> 
> Here is my reasoning...
> 
> If you have Pulsar A and Pulsar B, you have two points establishing a
> line.
>  You know the distance from A to B as well, since before heading out,
> you
> established A's distance from Earth, and B's distance from Earth, and
> thus,
> A's distance from B.
> 
> >From point C, you are at the unknown location.  All you can get at C
> at
> this point in time, is the bearing to A.  This gives you angular
> co-ordinates in this 3 dimensional triangle problem.	This bearing
would
> be
> described in both X and Y axis, but not in Z, because Z is unknown.
> 
> >From Point C, you are at the unknown location.  All you get at this
> point
> in time, is the bearing to B.  This gives you Angular co-ordinates for
> your
> X and Y, but not z because you don't know how far away you are from B.
> 
> Using the angles generated from your bearings to A and B, the only
> thing
> you don't have is distance in your equation right?  Hmmm, what about
> the
> distance from A to B which is a known quantity?  Now you have a
> triangle
> with *all* angles known, plus one side of the triangle's distance
> known.
> >From that, can you not determine your other two sides?  From that,
can
> you
> not determine your location in a rough manner?
> 
> I must be missing something.	Either that, or I am right, but am
> uncertain
> enough to say why...
> 
>		   Hal

Right, look; You need to get at least three bearings.

Bearing one gives you the line you lie on towards the object. This is
not 
useful. Because you don't know your own orientation, hence that line
could in 
fact be any line passing through the object.

Bearing two gives you a difference between those two bearings - a flat
angle 
between them. Assuming you know the location of the two objects, that
resolves 
you to being somewhere on a circle. (Note that taking the DIFFERENCE of
two 
bearings removes any dependence on your orientation). That circle
contains all 
the positions at which the difference of the bearings to the objects is
what 
you have. It's perpendicular to the line connecting the objects.

Bearing three, differenced with two, does the same thing, but gets a
different 
circle.

Unless you're really unlucky, those circles will only touch at one
point. 
That's you that is.

If they touch at more that one, congratulations. Pick another object.

Actually, they won't touch. They never do, due to measuring
inaccuracies. 
You're somewhere around their closest approach point.

You don't need to know your distance from the objects - that's why these

are "bearings" and not "positionings". Locating them on a celestial
sphere is 
enough, assuming, and this is a big assumption, that you pick objects
which are 
not too far away. I'd suggest picking the bright ones...

{If they're too far away, they won't move against your effective
celestial 
sphere as you move..}

Determining distance is possible, but a lot more inaccurate than just
taking 
bearings.

As an aside, I came across a reference somewhere to a new sky survey -
"2MASS will produce the following data products:  A digital atlas of the
sky 
comprising approximately 4 million 8´ × 16´ Atlas Images, having
about 4´´ 
spatial resolution in each of the three wavelength bands. 
A point source catalog containing accurate positions and fluxes for ~300

million stars and other unresolved objects. An extended source catalog 
containing positions and total magnitudes for more 
than 1,000,000 galaxies and other nebulae. "

Their intention is that this wide field survey will allow them to
calibrate the