Astropy [0] lives at the heart of most work. It has a Python interface, often backed by Fortran and C++ extension modules. If you use Astropy, you're indirectly using libraries like ERFA [6] and cfitsio [7] which are in C/Fortran.

I personally end up doing a lot of work that uses the HEALPix sky tesselation, so I use healpy [2] as well.

Openorb is perhaps a good example of a pure-Fortran package that I use quite frequently for orbit propagation [3].

In C, there's Rebound [4] (for N-body simulations) and ASSIST [5] (which extends Rebound to use JPL's pre-calculated positions of major perturbers, and expands the force model to account for general relativity).

There are many more, these are just ones that come to mind from frequent usage in the last few months.

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[0] https://www.astropy.org/

[1] https://healpix.jpl.nasa.gov/

[2] https://healpy.readthedocs.io/en/latest/

[3] https://github.com/oorb/oorb

[4] https://rebound.readthedocs.io/en/latest/

[5] https://github.com/matthewholman/assist

[6] https://github.com/liberfa/erfa

[7] https://heasarc.gsfc.nasa.gov/fitsio/