Escape velocity

As cosmic speeds velocity values ​​are referred to that have a special meaning in the celestial mechanics. In the simplest case, the speed will be described, which is necessary for a stable orbit to achieve a central body.

First cosmic velocity or circular velocity

When a body moves listless on a circular path around the center of a planet (or other celestial body ) around, the gravity of the planet causes the centripetal acceleration

It is the gravitational constant, the distance between the missile and the center of the planet, the mass of the planets and the velocity of the missile on the circular path. The circular velocity is obtained by rearranging the above equation to

The first cosmic velocity is that circular velocity that allows a body directly on the surface of a planet around this to revolve around. Illustratively stated, she is the rate at which a horizontal shot cannonball would circle the earth without touching the ground. Since the distance to the center of the planet, in this case the radius R corresponds to the planet is set.

The first cosmic velocity for the Earth is about 7.91 km / s ( 28476 km / h ). For the moon it is 1.68 km / s ( 6048 km / h ). The orbital period of this orbit corresponds to the Schuler period.

For each circular movement of a certain amount, the circular velocity is always less than. Satellites have a web height of at least 150 km above the Earth's surface because of friction with the Earth's upper atmosphere slows down too much otherwise the satellites. At 150 km altitude the necessary path velocity of the satellite is 7.815 km / s

A portion of the required circular velocity is already applied at the start of the Earth by the Earth's rotation, depending on the latitude of the starting location and the starting direction. In the ideal case ( starting at the equator to the east ), this contribution is about 460 m / s

Second cosmic velocity or escape velocity

The second cosmic velocity is the minimum speed for an open, non- returning path. At this rate, the kinetic energy of the specimen equal to its bonding energy in the gravitational field. On the surface of a spherical celestial body is thus

Changing according to results

The gravitational acceleration of the celestial body is. The escape velocity by a factor greater than the first cosmic velocity. At constant average density, the escape velocity scales linearly with. The table provides examples.

Geometrical Meaning

If a missile, which is located on a circular orbit around a planet, gets a speed boost in the flight direction, then its trajectory is deformed into an ellipse. , The speed is further increased, the eccentricity of the ellipse is increased. This continues until the far point of the ellipse is infinitely far away. From this speed, the body is no longer in a closed path, but the ellipse opens to a parabolic path. This will happen when the missile reaches the second cosmic velocity.

While the body away from the planet, it is by its gravitational further decelerated, so that it stops only at an infinite distance. If, however, the second cosmic speed is exceeded, the trajectory takes the form of a hyperbola - Asts - in this case a speed called the hyperbolic excess speed or hyperbolic excess speed and characterizes the energy of the hyperbolic orbit remains at infinity. It is calculated from the sum of the energies, ie, the squares of the velocities, analogous to the calculation in the following section. Also common to specify the square of the velocity ( ie energy per mass ) frequently c3, with the formula sign.

Escape velocity in space

Interplanetary space probes often are first on an Earth orbit ( parking orbit ) before the engines will be ignited again and the missile to accelerate to the required speed faster than the escape velocity (see below). Here, the rotational speed of the earth around the sun makes an appropriate choice of the trajectory already a large contribution to the necessary terminal velocity.

For trajectories to the moon the escape velocity does not need to be fully achieved, but the maximum distance of the missile to the ground the distance Earth-Moon must comply. The actual trajectory is not algebraically computable by the influence of the moon ( restricted three-body problem ).

Black Holes

In the extreme case corresponds to the escape velocity of the speed of light, so that it can leave the gravitational field at a certain distance not even light. This is the case for a black hole ( a notion that was first studied in 1796 by Pierre Simon Laplace ). Here is

Which for the critical radius

Obtained and is referred to as a black plate radius. This can also write the escape velocity as

This formula shows that the escape velocity from the ratio of the radius depends on the Schwarzschild radius. If the radius is small and approaches the Schwarzschild radius, the escape velocity increases, the two radii are equal, it reaches the speed of light. This is about the collapse of a neutron star into a black hole.

Third cosmic velocity

The third cosmic velocity is the escape velocity from the Sun, calculated from the Earth's orbit from. For this use again the formula for the second cosmic velocity, where now the mass of the Sun and the distance between Earth and Sun are used.

This gives v3 = 42.1 km / s

The above escape velocity is only valid for a body that moves away in the distance of the earth from the sun. But when a rocket from the earth and starts to leave the solar system, they must overcome the common gravitational field of the Earth and Sun. For the start of the earth you can make the rotational speed of the earth around the sun advantage, since they are already is 29.8 km / s. Thus, in a firing direction tangential to the Earth's orbit is only one speed of DELTA.v = 42.1 km / s - 29,8 km / s = 12.3 km / s relative to the earth necessary to leave the gravitational field of the sun alone. Then the body needs in addition the escape velocity of the earth.

For the necessary speed, this results in a total

.

Not considered here is the rotation speed of the earth. This one can also make advantage and is the reason why many spaceports are as close to the equator. Namely, there is the rotational speed of the earth at the maximum.

Fourth cosmic speed

The fourth cosmic velocity is finally necessary to leave the galaxy. The indication that the sun needed for one orbit around the galactic center at a distance of 28,000 light years, 230 million years, results from the third Kepler 's law for the inner mass of the galaxy, so that the escape velocity of the Quiet Sun from about 320 km / s. If one uses the rotation speed of the Sun around the galaxy, the / s at 220 km, the fourth cosmic speed reduced to about 100 km / s