Fig. 1. (a) Light ray trajectories of universal gravitation analogy and Schwarzschild precession analogy of the star S2’s orbit around the nearest massive black hole SgrA* candidate (in the origin). Trajectories start from the perihelion (0,−rp) (the cyan dot), and rp=a(1−e)=118.922Au. The red dashed elliptically closed curve is the analogy of universal gravitation. The blue curve varying with the “time” ζ is the mimicking of Schwarzschild precession. Per orbit of this precession is 12.1′, and here it is about 3° for 15 periods. The inset in the lower right corner is an enlarged view of the trajectories in the upper dashed white box. The background color map is the logarithmic refractive index distribution log(n) of the induced gradient lens mimicking Schwarzschild precession. The profile n(r) goes infinitely at the origin, and it equals 0 in the region outside r=2.063Au [the plotted minimum value log(n)=−3 here]. (b) and (c) The x components and y components of these two trajectories, respectively. Three periods are plotted, and the difference between universal gravitation and Schwarzschild precession is clearly shown in curves of x(ζ).

Fig. 2. Light ray trajectories in the lens n(r)=2(E+1r+λr2) with E=−12 induced by the modified Newton potential of two different λ. The color maps show the corresponding distributions of the refractive index profiles. All the rays start from point (1, 0), as indicated by the red dot. First row: λ=1; (a) and (b) correspond to the launching angles ψ=34π and ψ=13π, respectively. (a) The ray collapses into the center, like the trapping effect of a black hole. α equals 33i, which is a pure imaginary number. (b) The ray is closed after traveling around the origin three times, with α=13. Second row: λ=−637; (c) and (d) represent the launching angles ψ=16π and ψ=14π, respectively. (c) The trajectory is in precession and keeps dancing around the center, where α is 735. (d) The closed ray trajectory of seven rotational symmetric petals. It joins into the starting point after traveling around the origin five times, and α is 75.

Fig. 3. Light ray trajectories in the gradient lens n(r)=2(E−12r2+λr2) with E=1 induced by the modified Hooke potential of two different λ. The color maps and color bars indicate the refractive index profiles. Rays start from the point (1, 0) (in red dot). First row: λ=45422. (a) ψ=18π. Trapping effect of the center with α=i−1+45256csc2(π8). The ray first travels some distance towards the positive x direction and finally turns back to the center. (b) ψ=13π. Closed trajectory with seven petals. It travels around the center four times and 2α=74. Second row: λ=15(2−3)2(−34+153). (c) ψ=1718π, and α is an nonrational number herein. The orbit is in precession and restricted in an annular region. (d) ψ=112π. The closed ray path has only eight petals with 2α=8.