If the Sun were to suddenly disappear, the gravitational force that holds Earth in its orbit would vanish, and Earth would no longer be bound to the Sun. As a result, Earth would travel in a straight line with a constant velocity, tangential to its previous orbit.
The time it would take for Earth to completely leave its orbit would depend on its initial velocity and the distance from the Sun. The velocity of Earth in its orbit is approximately 30 kilometers per second (18.6 miles per second). However, the actual value can vary slightly depending on Earth's position in its elliptical orbit.
To estimate the time it would take for Earth to escape its orbit, we need to consider the concept of the orbital period. The orbital period is the time it takes for Earth to complete one revolution around the Sun, which is approximately 365.25 days or one year.
If we assume that Earth's velocity remains constant after the Sun disappears, we can use the formula:
Time = Distance / Velocity
The distance traveled by Earth in one orbit is the circumference of its orbit, which is given by:
Distance = 2 * π * Semi-Major Axis
The semi-major axis is the average distance between Earth and the Sun, which is approximately 149.6 million kilometers (93 million miles).
Using these values, we can calculate an approximate time:
Time = (2 * π * 149.6 million km) / (30 km/s) ≈ 314,000 seconds
Converting seconds to years:
Time ≈ 314,000 seconds / (60 seconds * 60 minutes * 24 hours * 365.25 days) ≈ 9.96 years
Therefore, it would take approximately 9.96 years for Earth to leave its orbit and travel in a straight line if the Sun were to suddenly disappear. It's important to note that this estimation assumes a constant velocity and neglects any gravitational effects from other celestial bodies, which could slightly alter the trajectory and time.