[Spice_discussion] question concerning apparent velocities &light-time

[John Irwin]

Ian,

First of all, I don't think your treatment for the transmission case is
correct. In this situation we're referring to the advanced time t' = t + LT
rather than the retarded time (assuming LT is an unsigned quantity: LT =
|r(t)|/c). So the calculation involves finding the position and velocity of
the target when it receives the signal from the "observer".

We can generalise the derivation using an appropriate sign (s) which takes
the value +1 for the transmission case and -1 for the reception case. Then
by definition

   LT = s*(t' - t) = |R2(t') - R1(t)|/c

where t is the time at the observer (when it receives a signal from the
target in the reception case or when it transmits a signal to the target in
the transmission case) and t' is the time at the target (when it transmits a
signal to the observer in the reception case or when it receives a signal
from the observer in the transmission case).

Differentiation of the second equality with respect to t leads to

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

The apparent velocity of the target is then calculated from

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

We can also show from the first equality that

   dt'/dt = 1 + s*dLT/dt

which means the correction factor for the transmission case is 1 + dLT/dt,
rather than the general 1 - dLT/dt you had assumed before.

All this is pretty much how SPICE describes and implements the light-time
problem in both reception and transmission.

You also ask about using the Lorentz transform to make a higher-order
calculation of the apparent velocity. I'm afraid that's something I've
neither attempted nor understood sufficiently well to be of any use. But one
thing I can say is that I don't think you can apply it directly to the
velocities because it is only a coordinate transformation (unless I've
misunderstood your reference to an equation in Jackson's book, which I do
not have at hand).

However, we have only been talking about the light-time problem so far, and
if you're not fussed about gravitational effects like the bending of the
light path and the differential clock rates, then I would think the
calculation we have covered already is accurate enough.

There is also, of course, the need to include stellar aberration, and to
treat it as a relativistic effect if you need second-order accuracy. The
main effect of stellar aberration is to rotate the direction of the light
signal received by or transmitted from the observer. This rotates the
apparent position of the target as well as its apparent velocity. However,
although the components of the apparent velocity change, the radial velocity
is preserved, so if you're only interested in the radial velocity, stellar
aberration has no effect.

Let me know if you still want the results of my derivation of the apparent
velocity due to the effect of relativistic stellar aberration. It's quite a
messy calculation and I would need to copy it from my notes. Although you
could try it yourself by differentiation of the "standard" expression for
calculating the aberrated direction.

John. 

> -----Original Message-----
> From: Ian Avruch [mailto:i.avruch at sron.nl] 
> Sent: Wed 2011-Jan-19 02:25 PM
> To: John Irwin; Nat Bachman
> Cc: spice_discussion at naif.jpl.nasa.gov
> Subject: Re: [Spice_discussion] question concerning apparent 
> velocities &light-time
> 
> Dear Nat & John,
> 
> Thanks for responding so quickly! I understand your 
> explanation, and I 
> can derive the
> the same expression John gives for the term (1 - d(LT)/dt) in the 
> transmission case, namely:
> 
> t     =  ephemeris time
> LT(t) = transmission light-time from 2 to 1 at time t
> t'    = t - LT
> R1(t)       =  ephemeris coordinates of observer
> R2(t')      =  LT-corrected ephemeris coordinates of target
> V1(t) = dR1/dt  = ephemeris coordinate velocity of observer
> V2(t')= dR2/dt' = ephemeris coordinate velocity of target at t-LT
> 
> r(t) = R2(t') - R1(t)
> n = r/|r|
> 
>       (this is a vector equation)
> 
> dr/dt = dR2(t')/dt' * dt'/dt - dR1/dt
>       = V2(t') * (1 - d(LT(t))/dt) - V1(t)
> 
>       = V2(t') * ( 1 + n.V1(t)/c)  - V1(t)  
>                  ---------------               (1)
>                  ( 1 + n.V2(t')/c)
> 
> 
> The error in neglecting the term dt'/dt relative to keeping it is of
> order, for small velocities,
> 
> delta(dr/dt) ~ V2(t') * (|V1|/c + |V2|/c)      (2)
> 
> In fact I'm interested in including Special Relativity in calculating
> the apparent velocity of 2 to 1, because I want apparent radial
> velocities in the solar system to be 'correct' to v/c <~ 1e-8, which
> requires terms of order beta**2 in the Lorentz transform.
> 
> The way I would like to do it is to take the retarded velocity V2(t')
> as an observable in the SSB frame, then boost it to the frame 
> of observer 1
> with a Lorentz transform
> 
> V21 = A_boost(V1) * V2
> 
> Were the V's are now 4-vectors and
> A_boost(V1) is the boost matrix as in Jackson "Classical 
> Dynamics" eqn 
> 11.98 .
> I reckon I can ignore acceleration over the ~10 seconds of interest,
> and also GR effects on the clock rates at 1 and 2 and on the 
> light-time.
> 
> I've spilt some ink over the algebra but still don't have a vector
> expression for V12 I trust; too bad, I would have liked to 
> include here the
> difference with the SPICE result, as in equation (2). But 
> must move on...
> 
> So, if you don't mind, two further questions:
> 
> (a) Do you think it's valid to simply take V2(t') and V1(t) as
>     two SSB-frame velocities, and then ask what velocity 1 sees for 2
>     via a Lorentz transform?  The important fact is my 
> desired accuracy in
>     V_radial/C ~ 1e-8, by which I can ignore accelerations & GR.
>     I'm worried about forgotten factors; the d(LT)/dt has spooked me.
> 
> (b) Do you have the relativistic vector expression for V21, the
>     apparent velocity of 2 wrt 1, which I can check (steal)?
> 
> Cheers
> -Ian