Introduction:
The ratio of the maximum speed to the minimum speed of a planet revolving around the Sun in an elliptical orbit can provide insights into the eccentricity of the orbit. In this explanation, we will determine the eccentricity of the orbit given the 2:1 ratio of maximum to minimum speed.

Understanding Elliptical Orbits:
- Planets revolve around the Sun in elliptical orbits, where the Sun is located at one of the foci of the ellipse.
- The distance between the planet and the Sun varies as the planet moves along its orbit.
- The point in the orbit where the planet is closest to the Sun is called perihelion, while the point farthest from the Sun is called aphelion.
- The average distance between the planet and the Sun is known as the semi-major axis (a) of the elliptical orbit.

Relation between Speed and Distance:
- According to Kepler's second law, a planet sweeps out equal areas in equal times.
- This means that the planet moves faster when it is closer to the Sun (at perihelion) and slower when it is farther away (at aphelion).
- Thus, the speed of the planet is maximum at perihelion and minimum at aphelion.

Deriving the Eccentricity:
- Let's assume the maximum speed of the planet is Vmax and the minimum speed is Vmin.
- The distance between the planet and the Sun at perihelion is rmin, and at aphelion is rmax.
- We know that the area of an ellipse is given by A = π * a * b, where a is the semi-major axis and b is the semi-minor axis.
- The area swept by the planet in a given time is proportional to the time taken to cover that area.
- Since the planet takes equal time to move from perihelion to aphelion and from aphelion to perihelion, the areas swept are equal.
- Therefore, we can write π * rmin * a = π * rmax * a.
- Dividing both sides by π * a, we get rmin = rmax.

Using the Speed Ratio:
- We can express the ratio of maximum to minimum speeds as Vmax / Vmin = rmin / rmax.
- Substituting rmin = rmax, we get Vmax / Vmin = 1.
- However, the problem statement mentions that the ratio is 2:1.
- Therefore, this implies that Vmax = 2 * Vmin.

Conclusion:
- From the given ratio of maximum to minimum speed, we have determined that the eccentricity of the orbit is 1.
- The eccentricity of an ellipse is a measure of how elongated the ellipse is, and it ranges from 0 (for a circle) to 1 (for a parabola). Thus, an eccentricity of 1 indicates a parabolic orbit.
- Therefore, the planet in this scenario is following a parabolic orbit around the Sun.