[Spice_discussion] question concerning apparent velocities &light-time

[Ian Avruch]

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


 

John Irwin wrote:
> 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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>>     
>
>   

-- 
Ian Avruch  SRON/Kapteyn Institute
            Postbus 800
            9700 AV GRONINGEN
            (0031 | 0) 50 363 8759