To cause Earth to fall out of its orbit around the Sun, a substantial amount of mass would need to be added. The gravitational force between two objects depends on their masses and the distance between them, as described by Newton's law of universal gravitation.
Adding mass to Earth would increase its gravitational pull on the Sun, causing a change in the dynamics of the Earth-Sun system. The additional mass would result in a stronger gravitational attraction between Earth and the Sun, altering the balance between gravitational force and the centrifugal force that keeps Earth in orbit.
To calculate the exact amount of mass required to cause Earth to fall out of its orbit, a detailed analysis of the system's dynamics would be necessary, considering factors such as the new mass distribution, the distance between Earth and the Sun, and the current orbital velocity. It is a complex problem that involves numerical simulations and gravitational calculations.
However, it is worth noting that Earth has a very large mass already (approximately 5.97 x 10^24 kilograms), and adding a significant amount of mass to Earth would likely have other catastrophic effects on the planet, such as drastic changes in its structure, atmosphere, and overall stability.