Escape velocity. Keplerian motion

Let us go back to the different types of orbital motions. Suppose, we are launching a spacecraft from Earth. The shape and size of the spacecraft will depend on its initial velocity (here we will ignore the mass of the spacecraft since it is much smaller than the mass of the Earth). If the velocity is too low, the craft will start to describe an elliptic orbit, but its size will not be large enough and it will crash on Earth. If we increase the velocity until it reaches a special value of v = (GM/r)1/2 (also called circular velocity, M is the mass of the Earth), the spacecraft will obtain a circular orbit with a radius r around the Earth. The radius of the orbit r is calculated from the centre of the Earth, and the eccentricity of the orbit will be zero.

If we increase the velocity above the circular, the spacecraft will reach an elliptical orbit around the Earth. The higher the velocity value goes, the more eccentric the orbit will be. Until it reaches another specific (critical) value (v = (2GM/r)1/2), at which the elliptic orbit will “tear apart” to form a parabola. If we keep increasing the velocity above the parabolic value, the parabola will transform into hyperbola and again, the higher the velocity, the more spread out will be the arms of the hyperbola.

In all the above cases, the velocity is always inversely proportional to the square root of the distance. Motion with such velocities is called Kelperian motion.

When applied to the Solar system planets, this law of motion indicates that the more distant from the Sun the planet is, the slower its orbital motion will be. For example, Mercury will have the highest orbital velocity (about 47 km/s on average, i.e. its orbital period will be about 88 days), Neptune will have the lowest velocity (5.5 km/s on average, and orbital period of 165 years).

Let us go back to our spacecraft. If its orbit is elliptic, it is gravitationally bound to the Earth (will become its satellite), and the velocity is called orbital velocity (circular velocity) relative to the Earth.

If the orbit is parabolic or hyperbolic, the spacecraft will be able to “escape” Earth’s gravitational pull. The necessary velocity is called escape velocity. For a star, or a planet, the escape velocity is v = (2GM/r)1/2, i.e. the one, needed to obtain a parabolic orbit. In the general case, where G is the gravitational constant, M is the mass of the body, whose gravitation the object escapes, and r is the distance from the centre of mass to the escaping object. For an object, launched from the Earth’s surface, the average value of the escape velocity is 11.2 km/s.

Another escape velocity will be one, needed to escape the gravitational pull of the Sun. The formula is the same, but M will be the mass of the Sun, and r – the distance Sun – Earth (if it’s a craft launched from Earth).