Shapiro test interstellar/intergalactic

Question

I understand that the Shapiro time delay effect is one of the four classic solar system GR tests.

I understand the EM waves have to pop back from a planet beyond the star and then pass next to the sun coming backwards.

Question:

1. has anybody ever done the same test interstellar, so with another solar system? Alpha Centauri is only 4.37 light years away, so it sounds doable. (It might be harder to time for the planet orbitals in that solar system, but it sounds still doable, just like with the Shapiro effect, the EM wave has to pop back from a planet from beyond Alpha Centaury, we know the planets' orbit in that solar system too, so we can calculate the right position when the EM wave gets there to pop back from it.)

Answer 1

In principle yes, but I don't think that the Shapiro time delay is measurable interstellar/intergalactic. To do this a radar signal would have to pass a potential well (a Star or a galaxy) and then to be reflected back to the source and detected there. Both, reflection and detection don't seem to be within the scope of present technology.

Answer 2

The Shapiro delay has already been experimentally verified and used (see two examples at the end of this answer).

Let us assume a radio beam between $ A $ and $ B $ at radial coordinates $ r_A $ and $ r_B $ respectively from a massive object of mass $ M $, with the massive object "between" $ A $ and $ B $, and $ r_0 $ the minimum distance of the radio beam to the center of the massive object.

The two-way duration of the radio beam according to Newton theory is: $$ T_{Newt}=\frac{2}{c}\left(\sqrt{r_A^2-r_0^2}+\sqrt{r_B^2-r_0^2}\right) $$ with $ c $ speed of the light in vacuum.

The two-way duration actually measured $ T $ is always greater than $ T_{Newt} $ and after a few calculations, whereas $ r_0\ll r_A $ and $ r_0\ll r_B $, the delay $ \Delta T $ - called Shapiro delay or light delay - can be defined as: $$ T-T_{Newt}=\Delta T\simeq\frac{4GM}{c^3}\left[\ln{\left(\frac{4r_Ar_B}{r_0^2}\right)+1}\right]\ \ \ \ \ [1]$$ with $ G $ gravitational constant and $ M $ mass of the massive object.

Numerical examples:

$ [a] $ for a Mars probe, the maximum delay is around $ 0.24\ ms$. Thanks to the Cassini probe, it was shown in 2003 that the delay of light is consistent with general relativity (i.e. formula $ [1] $) to within $ 2\ 10^{-5} $.

$ [b] $ in 2010, the Shapiro effect was used to precisely measure the mass of a neutron star in a binary system with a white dwarf. The neutron star is observed as a pulsar. Once per orbit, the white dwarf passes in front of the neutron star and we then observe a delay of $ 25\ \mu s $ in arrival time of the pulses, which can be interpreted as the Shapiro effect effect due to the gravitational field of the white dwarf. From this, we deduce the mass of the white dwarf and, via the orbital parameters, that of the neutron star: $ M=1.97\pm 0.04 $ mass of the Sun.

Hoping to have answered the question,

Best regards.