## 5 Summary  
The theory of the detection of an extra-solar planet using a rotational shearing interferometer has been expanded to the case when this instrument is not precisely aligned on the star. For the small amounts of misalignment, the planet signal modulates the constant star signal. This finding was previously demonstrated for the case of the perfect alignment. For the relatively large displacement of the star from the origin, corresponding to the displacement equal to the Mercury orbit of 0.4 AU, some small variation in incidence may be detected with a high bit depth detector and subtracted. The relative change in incidence of the star fringe pattern due to the star displacement or misalignment may be estimated to be of the order of magnitude of the square of the misalignment angle in radians.  
An interesting trend is noted that the increasing number of star fringes is generated when the distance of the star center increases from the coordinate origin. Similarly, as the displacement of the star center becomes smaller than the radius of the Mercury orbit, the red-giant star generated less than 0.5 fringe. The Mercury orbit could be considered as the limiting distance of the star center that still produces an identifiable fringe. It represents a large error in the star position, so one may safely conclude that for relatively small errors, a negligible gradient in the star incidence is observed. The interference pattern of the planet thus dominates the sum of the planet and the star incidence, with the star background contributing a small incidence gradient. The background incidence gradient decreases with the decreasing alignment error.  
Alignment errors corresponding to the Mercury orbit are not expected in a process of bore-sighting on the star. If one accepts as reasonable errors those equal to the radius of a normal, main sequence star like our Sun, such errors do not result in discernable incidence changes. When a red giant star is considered and its potential displacement to the edge of its radius (nearly the Earth orbit), this indeed represents a significant error in alignment and would result in a formation of a star generated fringe. Then the image processing techniques may be used to separate the star incidence contribution from that of the planet.  
For most star alignment errors appreciably smaller than the radius of the Mercury orbit, even in the case of the red giants, the detected incidence simplifies to a constant background term and a sinusoidal modulation. The modulation is only present when there is a planet in the field of view. Furthermore, the fringe separation may be changed by changing the orientation of the Dove prism to verify the signal veracity.  
A large star misalignment corresponds to a star center location that is larger than the radius of the Martian orbit. Such a case may arise when a displaced star is found in the same field as the star of interest located on the optical axis. One may observe the star equal-gray-level fringe along a specific angle, while the equal gray-level fringe due to the planet is oriented along a different angle. This angle increases as the planet orbits the star. Finally, the planet fringes change orientation with respect to the star fringes during the local year, again introducing a useful temporal dependence.  
The research described here was partially supported by the Air Force Office of Scientific Research, under contract No. (FA9550-18-1-0454).  
Not applicable.  
A portion of this work was presented at the virtual conference Advances in 3 OM Conference in Timisoara, Romania, December 2021.  
The author declares no conflict of interest.