[Spice_discussion] question concerning apparent velocities &light-time

[John Irwin]

Ian,

I have previously worked out all the details of the apparent velocity
calculation, for both the light-time problem and the stellar-aberration
problem, and for the newtonian and relativistic versions of the latter. So I
may be able to help.

There are no references that I can find, which is why I dove in myself, but
as Nat Bachman said, it's just calculus. However, I have only done the
derivations for the reception case (I have no use for the transmission
case), but the principles are the same.

Now, when you mention the "relativistic addition of velocities", that sounds
like you're considering the stellar-aberration problem, not the light-time
problem. The factor of which you speak, (1 - d(LT)/d(t_eph)), is only a
light-time effect. This factor is strictly non-relativistic too, as Nat
warned, as it is based on the cartesian separation of the target from the
observer.

With that in mind, this factor is actually the rate of change of the
transmission time (t') with respect to the reception time (t). Given the
apparent, light-time corrected position of a target(index 2) relative to an
observer(index 1):

   r(t) =  R2(t') - R1(t)

where uppercase variables are absolute (barycentric say) and lowercase
variables are relative, then the apparent velocity is:

   v(t) = dR2(t')/dt - dR1(t)/dt

        = dR2(t')/dt'*(dt'/dt) - V1(t)

        = V2(t')*(dt'/dt) - V1(t)

Calling the light-time LT, then LT = t - t', and

   dLT/dt = d(t - t')/dt = 1 - dt'/dt

hence

   dt'/dt = 1 - dLT/dt.

Note that derivatives are with respect to the reception time as this is a
reception-time problem.

Working through the details, an expression can be found for dt'/dt assuming
the light-time LT = |r(t)|/c:

   dt'/dt = 1 + n(t).V1(t)/c
            -----------------
            1 + n(t).V2(t')/c

where n(t) is the unit vector along r(t) and the "." are scalar products.

The treatment for the stellar-aberration problem is a little more involved.
Let me know if you want to compare notes.

John. 

> -----Original Message-----
> From: spice_discussion-bounces at naif.jpl.nasa.gov 
> [mailto:spice_discussion-bounces at naif.jpl.nasa.gov] On Behalf 
> Of Ian Avruch
> Sent: Wed 2011-Jan-12 07:36 PM
> To: spice_discussion at naif.jpl.nasa.gov
> Subject: [Spice_discussion] question concerning apparent 
> velocities &light-time
> 
> Hi,
> According to the documentation, for example module SPKEZR, 
> since recently
> the light-time-corrected, apparent velocity includes a factor
> (1 - d(LT)/d(t_eph)).
> Can anyone provide a reference for this derivation?
> It doesn't match what I get starting from relativistic addition of 
> velocities,
> although I'm sure there needs to be a dilation term like that.
> 
> Sorry if this dredges up old discussions; I wasn't able to them if so.
> Thanks much,
> -Ian
> 
> -- 
> Ian Avruch  SRON/Kapteyn Institute
>             Postbus 800
>             9700 AV GRONINGEN
>             (0031 | 0) 50 363 8759
> 
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