I am doing some satellite-related research and would like to verify whether an assumption I am making about the satellite geometry is reasonable or not.
Essentially, a LEO (low-earth-orbit) satellite has a circular coverage onto the earth. How large the circle will be depends on the elvation angle between the satellite and the earth and the satellite altitude (for example the side view in the picture shows that $SAT_1$ has an elevation angle of $40$ degrees). Say if $SAT_1$ moves a little bit to new a location and we represents the new snapshot as $SAT_2$, and the distance between $SAT_2$ and $SAT_1$ is $\Delta x$. My question is, will the $SAT_2$'s cicular coverage also shifts by roughly $\Delta x$? More specifically, will a point on the $SAT_1$'s coverage to its counterpart on the $SAT_2$'s coverage be roughly $\Delta x$? I guess the side view and top view in the attached picture can help illustrate. My assumption is yes, but I would like to double check. (If there are some errors, I guess the errors are mainly from the curvature of the earth surface. how large will the error be?)
(Some context: I am doing some LEO satellite networking/communication-related topic. Specifically, given two snapshots of a satellite, I am trying to estimate how long the coverage at the old snapshot will move to the boundary of the new coverage, so the user at the boundary of the new coverage can start using the satellite networking. I guess if $\Delta x$ assumption is reasonably accurate, the time the user needs to wait is around $\frac{\Delta x}{\text{satellite travel speed}}$. I feel this is essentially a geometry problem, so I posted here.)
Thank you very much in advance!
Your approximation works if the satellite is relatively close to the surface of Earth compared to the radius of the Earth. Assuming that the Earth is a sphere and the orbit of the satellite is circular (i.e., the distance from the center is constant), if $SAT2-SAT1=\Delta x$ is small, then the shift of the covering is $$\Delta c=\frac{R}{R+h}\Delta x ,$$ where $R$ is the radius of Earth and $h$ is the altitude of the orbit of the satellite measured from the surface.
The quantity $\frac{R}{R+h}$ is just the ratio between the radius of Earth and the radius of the orbit of the satellite. The intuition is that the length of a circular arc is proportional to its radius if the angle is fixed.
If$\Delta x$ is not small, the formula still works, but you have to measure $\Delta x$ and $\Delta c$ as the length of the curve paths along the orbit of the satellite and the surface of the Earth respectively.