The Earth moves around the Sun in an elliptic trajectory due to gravity. If another Sun like star is brought near the Earth, what will be the shape of the trajectory ?

The Earth's current orbit around the Sun is a stable two-body system resulting in an elliptical trajectory [c2][t2]. However, introducing a second Sun-like star transforms the system into a 'three-body problem'. Unlike the two-body problem, which has closed-form solutions like ellipses or parabolas [t2][t4], the three-body problem is famously known for its chaotic and non-integrable nature. In such a system, the gravitational interactions become extremely complex and sensitive to initial conditions, making the long-term trajectory undeterministic and unpredictable [t5]. While simple linearized models work for two bodies, they break down in three-body environments where dynamics vary extremely over the trajectory [t5]. Therefore, the Earth would no longer follow a simple conic section like an ellipse or circle; its path would become chaotic and undeterministic depending on the specific masses and velocities involved.

Sources

1. [1] Physical Geography by PMF IAS, Manjunath Thamminidi, PMF IAS (1st ed.) > Chapter 19: The Motions of The Earth and Their Effects > Perihelion and Aphelion > p. 255