Use VLBI astrometric measurements along with coordinate system offset in the timing model

[abhisrkckl]

This is based on https://arxiv.org/abs/2509.21203, mostly for simulation purposes.

[abhisrkckl]

Example par file:

        RAJ         05:00:00    1
        DECJ        15:00:00    1
        POSEPOCH    55000
        VLBIAX      -0.15
        VLBIAY      0.05
        VLBIAZ      0.21
        F0          100         1
        F1          -1e-15      1
        PEPOCH      55000
        TZRMJD      55000
        TZRFRQ      1400
        TZRSITE     gbt
        UNITS       TDB
        EPHEM       DE440

The coordinate offsets are not fittable parameters since they cannot be determined from a single pulsar.

[abhisrkckl]

Relevant expressions from Sosa Fiscella et al:

[dlakaplan]

would be helpful to have a paper reference or a little more math to explain this

[sophiasosafiscella]

Hello @dlakaplan and @abhisrkckl ! If you need a paper reference or a little more math, I'd recommend Wang et al.'s paper. That being said, the notation wasn't super intuitive when I read it so please let me know if I can help with that.

[abhisrkckl]

I have added a reference to Madison et al, since I think that is the original paper where this equation is from.

[abhisrkckl]

Please let me know if there is an older referecne I should use.

[dlakaplan]

@mtlam : this looks good to me, but don't know if you have thoughts.

[sophiasosafiscella]

Relevant expressions from Sosa Fiscella et al:

I'd say that the real relevant equation is Eq. (7) from the paper:

I'd be happy to provide the script that already implements all this math, if that'd be of any help.

[mtlam]

I'm going to mostly defer to Sophia as the expert here. It looks like the goal is to allow for these kinds of offsets in simulated TOAs?

[abhisrkckl]

The main aim here is simulations, and to allow VLBI measurements to be used directly in par files.

Fitting for the coordinate offsets is not an aim, since that can't be done with single pulsars.

[abhisrkckl]

I'd say that the real relevant equation is Eq. (7) from the paper:

If I understand correctly, this equation is used for estimating the coordinate offset. However, this PR is not doing that. It is only for simulating the offset, and for using the offset in a par file if it's already known.

I am not adding the coordinate offset estimation part since it requires analyzing multiple pulsars together, and we had decided to keep PINT strictly single-pulsar to avoid overlapping functionalities with enterprise etc.