[fitswcs] Relativistic transverse Doppler effect [RESEND]

[Mark Calabretta]

Greetings,

When Eric announced the latest draft of Paper III on Feb/24 he mentioned
that I would make a suggestion regarding the introduction of a new
keyword to record the orientation angle for resolving the transverse
relativistic velocity.

Figure 1 of the paper shows that the transverse velocity potentially
has a very significant effect.  

To be concrete, I like to think of a relativistic jet emanating from a
distant galaxy.  In principle, the galaxy's systemic, cosmological
redshift can be measured separately and used to correct the jet's
observed redshift thereby providing its kinematic redshift (i.e.
associated with a true velocity) in the reference frame of the galaxy.

Clearly knowledge of the velocity is fundamental in studying jet
kinematics and dynamics.  However, it can only be computed from the
kinematic redshift if the jet's orientation angle is known.  Note that
the equations involving velocity in Table 3 are actually only correct
if the transverse velocity is zero.  For example, Eq. (12) is just a
special case of Eq. (2).  This is why the velocity is always referred
to in the paper as being "apparent".

There are instances where the orientation angle may be inferred by
geometry (e.g. by the observed tilt of an accretion disk) or by
modelling.  I therefore propose that the paper be augmented

   1) by the introduction of a new keyword to record the orientation
      angle, and

   2) by modifying the eight equations involving velocity in Table 3
      to take account of this orientation angle.

For example, Eq. (12) would become

                     1 + v sin(theta)/c
    lambda = lambda0 ------------------
                     sqrt(1 - v^2/c^2)

and its inverse, Eq. (16),

          L*lambda^2 - lambda0^2 sin(theta)
    v = c ----------------------------------- ,
            lambda^2 + lambda0^2 sin^2(theta)

where

    L = +/- sqrt{1 - (lambda0/lambda)^2 cos^2(theta)} .

theta is defined so that

  theta = -90 is towards the observer,
  theta =   0 is transverse, and
  theta = +90 is away from the observer,

and thus

    v_r = v sin(theta)          ...radial velocity component
    v_t = v cos(theta)          ...transverse velocity component
    v   = sqrt(v_r^2 + v_t^2)

Equations (8,9,12-17) would need to be changed, and unfortunately the
derivatives would probably be rather messy.

Mark Calabretta
ATNF