Determining The Inclination Angle of A Stellar Binary System

[Phys12]

I am a Physics undergraduate at the University of Texas at Arlington. I am currently taking an Astrophysics class in which my professor talked about the inclination angle of the orbital plane of the binary star system and how it is impossible to determine that angle. I was wondering why is it the case? What exactly makes it impossible to find out the inclination angle of a stellar binary system? I asked my professor and he said that he doesn't think my method would work because our instruments are not sensitive enough and recommended that I contact other people/professors and see what they say since this is not his area of research.

I tried playing around with some equations hoping to come up with a way of figuring out the inclination angle to get the true velocity of the star. I have attached what I have been able to come up with so far, however, I am not sure how feasible this would be.

 

[mfb]

One of the most sensitive spectrometers in the world, the one installed in
the Very Large Telescope in the European Space Observatory has a resolution
of 10⁵

That is the ability to resolve different spectral lines, to determine the radial velocity you "only" have to find the center, that can be done with a much higher accuracy. 1 m/s radial velocity measurements are possible with bright stars, a bit better is possible under ideal conditions with a lot of observation time (e. g. Alpha Centauri Bb - 0.5 m/s).

Let's consider a very close binary star with 200 km/s radial velocity and give it a distance of 10 parsec to make optical resolution difficult. The angle changes by 1/10 of an arcsecond, that changes the radial velocity by something like 500 km/s * sin(1/10 arcsecond) = 0.1 m/s. Maybe a factor 2 more if the alignment of the systems is ideal. Still too small to resolve it with current telescopes. HIRES at the ELT might be able to do it. It is focused on detecting planets however, not tiny changes in rapidly orbiting binary stars.
Closer binaries in even closer orbits would help as well, but I'm not sure if they exist. Overall the best future observatories could maybe get a very rough idea of the orientation of a few systems with ideal conditions if you spend a lot of observation time on it. I'm not sure how useful that is.

 

[Phys12]

That is the ability to resolve different spectral lines, to determine the radial velocity you "only" have to find the center, that can be done with a much higher accuracy. 1 m/s radial velocity measurements are possible with bright stars, a bit better is possible under ideal conditions with a lot of observation time (e. g. Alpha Centauri Bb - 0.5 m/s).

Let's consider a very close binary star with 200 km/s radial velocity and give it a distance of 10 parsec to make optical resolution difficult. The angle changes by 1/10 of an arcsecond, that changes the radial velocity by something like 500 km/s * sin(1/10 arcsecond) = 0.1 m/s. Maybe a factor 2 more if the alignment of the systems is ideal. Still too small to resolve it with current telescopes. HIRES at the ELT might be able to do it. It is focused on detecting planets however, not tiny changes in rapidly orbiting binary stars.
Closer binaries in even closer orbits would help as well, but I'm not sure if they exist. Overall the best future observatories could maybe get a very rough idea of the orientation of a few systems with ideal conditions if you spend a lot of observation time on it. I'm not sure how useful that is.

So, if it's the case that the only limitation in figuring out the inclination angle is the sensitivity of our detectors, then why is it said that it's impossible to figure out the inclination angle? My method seems easy enough that someone would've thought of it before me. When they say impossible, do they mean impossible in terms of our technological limitations?

 

[mfb]

When they say impossible, do they mean impossible in terms of our technological limitations?

Yes.
If you ignore technological limitations you can send a probe there.