The reference system
====================

.. figure:: images/ref-7.png
.. figure:: images/ref-6.png
   
   Left) Hazel reference system, rigth) diagram indicating the
   position of the LOS vector. [fig:figure1]

As we can see in Fig. [fig:figure1], the angle :math:`\chi` is measured
from :math:`X` to :math:`Y`, and :math:`\theta` is measured from
:math:`Z` to :math:`\Omega`. We can choose the angle :math:`\chi` in
order to simplify the equations. If we choose :math:`\chi= 0`, then
:math:`\Omega` is between :math:`Z` and :math:`X`. This configuration
only happens when :math:`X` is radial and points towards the disk center
(DC). However, if we choose :math:`\chi= 180`, then :math:`\Omega` is
between :math:`Z` and :math:`-X`. This configuration only happens when
:math:`X` is radial and points away from the disk center (DC). In the
rigth panel we see where is :math:`\Omega` if we choose the value of
:math:`\chi`.

The consecuences of choosing one option are: the reference system itself
and the equations to find all the posible solutions.

Example 1
----------

We have the following data from our observation and we choose
:math:`\chi=`\ 180d:

-  Position [arcsec]: :math:`x=-300.0`; :math:`y=-200.0` &
   Q\ :math:`>`\ 0: N-S

We can calculate the angle from the equator:

::

    alpha = np.arctan(y/x)*180./(np.pi) = 33.7d

Then we can calculate the heliocentric angle:

::

    theta = np.arcsin(np.sqrt(x**2.+y**2.)/960.)*180/np.pi = 22.1d

Now we can calculate :math:`\gamma` (must be measured from X to Y,
anticlockwise). In Fig. [fig:example1] (right) you have two solutions:
the purple and the blue one:

::

    Gamma(purple) = 360-(90+33.7)=236.3d    or     Gamma(blue) =(90-33.7) = 56.3

Q\ :math:`>`\ 0 is a line, not a direction. Gamma is defined [0,180º]
(look the Hazel GUI) so in principle you can choose the direction of
Q\ :math:`>`\ 0 which makes Gamma inside the range. In order to check
the result, you can execute the 3D plot to visualize the result (Fig.
[fig:example1]).

.. figure:: images/ref-4.png
.. figure:: images/ref-5.png
.. figure:: images/ref-3.png
Hazel reference system of the first example [fig:example1]

Then, the angles for this observation are:

::

    theta_OBS = 22.1d
    chi_OBS = 180.0d
    Gamma_OBS =(90-33.7) = 56.3

Example 2
----------

We have the same observation but we choose :math:`\chi=`\ 0d. Now,
:math:`\gamma` is again measured from :math:`X`, and it is the same as
before (Fig. [fig:example2]).

::

    Gamma(blue) =(90-33.7) = 56.3

.. figure:: images/ref-1.png
.. figure:: images/ref-2.png
.. figure:: images/ref-0.png
Hazel reference system of the second example [fig:example2]

Then, the angles for this observation are:

::

    theta_OBS = 22.1d
    chi_OBS = 0.0d
    Gamma_OBS =(90-33.7) = 56.3