  The Kerr-Sen black hole is a rotating charged black hole solution arising
from heterotic string theory. In 4-dimensions effective theory the bosonic
fields are: a $U(1)$ gauge boson, a Kalb-Ramond 3-form which is equivalent to a
pseudoscalar axion in 4-dimensions, the dilaton and the graviton. The coupling
constants in the theory are $\alpha^{\prime}$ (inverse string tension) and
$\kappa$ (inverse reduced Planck mass in 4-dimensions) and the charge of the
$U(1)$ field and the axion-photon coupling are related to these two. Sen found
a black hole solution (the Kerr-Sen black hole) with these fields as the
external hair of the black hole. In this paper we investigate the possibility
of determining the Sen solution from observations. The observations which can
test the Kerr-Sen black hole are: (a) determination of the shape of the photon
shadow, and (b) the rotation of polarization of photon due to axion hair. The
deviation from circularity gives the $U(1)$ charge of the black hole and
identification of this charge in terms of the photon coupling leads to a
prediction of frequency independent "Faraday rotation" in terms of black hole
parameters already determined from the shadow. Similar measurements of
Kerr-Newman black hole with axion hair have no correlation between the shape of
the image and the amount of "Faraday rotation". This correlation can be a
distinctive test of the Sen metric. In the recent observation from EHT of M87*
shadow, the deviation from circularity has an upper bound of 10%. If this
observation is refined to 1% accuracy then a definitive prediction of the
charge of the Kerr-Sen black hole and the "Faraday rotation" can be made.
Interestingly observations of "Faraday rotation" have shown that the effect is
independent of frequency pointing to an axionic hair interpretation for the
effect.
