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On 2018-04-03 12:30, John D. Howard wrote:
> If the speed of light was important for navigation almanacs how would you 
figure the position of a star that is 2000 light-years away?

It doesn't require any allowance for light time because the coordinates
in a star catalog are the place where the star was, when it emitted the
light that reached the solar system barycenter (center of gravity) at
some standard epoch (usually 2000 Jan 1). I.e., light time is already
included.

To be strictly correct, Earth is 8 light minutes from the barycenter, so
there is a small error due to the difference in light time. This is
called the "Roemer effect." The worst geometry occurs when the
barycenter, Earth, and star are in syzygy. In that case the error equals
the proper motion of the star in 8 minutes. It's a fraction of a
millisecond of arc even for the stars with the highest proper motions,
so normally we ignore light time.

That won't do for solar system objects. Compared to the stars their
angular rates are extremely high, so the light time correction is
sensitive to the observer's location. Light time is not included in the
JPL ephemerides. They give the barycentric geometric place of the body:
where it actually is, with respect to the barycenter. The user must
compute the astrometric place: where it was, when it emitted the light
that reaches the observer at the time of interest.

Suppose you want the geocentric astrometric place of Jupiter at time t.
Initialize tau (light time) to zero. From the ephemeris get the
barycentric geometric place of Earth at t and Jupiter at t-tau. Call
these quantities Eb(t) and Jb(t-tau).

Compute the first approximation of the astrometric place: J1(t) =
Jb(t-tau) - Eb(t). (This is identical to the geocentric geometric place.)

Now the iteration begins. Compute a more accurate light time: tau =
|J1(t)| / c, where c is the speed of light. With the new tau, compute a
more accurate astrometric place: J2(t) = Jb(t-tau) - Eb(t). Compare the
old and new astrometric places: J1(t) vs. J2(t). If they are equal, plus
or minus the desired accuracy, J2(t) is the astrometric place.
Otherwise, J1(t) = J2(t) and repeat the paragraph.

The more rigorous procedure in section B of the Astronomical Almanac
includes a light time correction due to the Sun's gravitation. This is
separate from the deflection of light, which is one of the corrections
in the transformation from astrometric place to apparent place.
Retardation and deflection are relativistic effects and insignificant at
Nautical Almanac precision.

   

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