Escape Velocity

If an object moves fast enough, it can escape from the gravitational field of the central body (to be precise: the field extends to infinity, so the object never really escapes, but is able to recede without any limit). If the escaping object has the minimum velocity allowing escape, it will have lost all its velocity at infinity (Fig. 6.11). There its kinetic energy is zero, since v = 0, and the potential energy is also zero, since the distance r is infinite. At infinite distance the total energy as well as the energy integral h are zero. The law of conservation of energy gives, then:

where R is the initial distance at which the object is moving with velocity v. From this we can solve the escape velocity:

For example on the surface of the Earth, ve is about 11 km/s (if m2 ^ m ®).

The escape velocity can also be expressed using the orbital velocity of a circular orbit. The orbital period P as a function of the radius R of the orbit and the orbital velocity vc is

2n R

Fig. 6.11. A projectile is shot horizontally from a mountain on an atmo-sphereless planet. If the initial velocity is small, the orbit is an ellipse whose pericentre is inside the planet, and the projectile will hit the surface of the planet. When the velocity is increased, the pericentre moves outside the planet. When the initial velocity is vc, the orbit is circular. If the velocity is increased further, the eccentricity of the orbit grows again and the pericentre is at the height of the cannon. The apocen-tre moves further away until the orbit becomes parabolic when the initial velocity is ve. With even higher velocities, the orbit becomes hyperbolic

Substitution into Kepler's third law yields 4n 2 R2 4n 2 R3

From this we can solve the velocity vc in a circular orbit of radius R :

Comparing this with the expression (6.41) of the escape velocity, we see that

6.10 Virial Theorem

If a system consists of more than two objects, the equations of motion cannot in general be solved analytically (Fig. 6.12). Given some initial values, the orbits can, of course, be found by numerical integration, but this does not tell us anything about the general properties of all possible orbits. The only integration constants available for an arbitrary system are the total momentum, angular momentum and energy. In addition to these, it is possible to derive certain statistical results, like the virial theorem. It concerns time averages only, but does not say anything about the actual state of the system at some specified moment.

Suppose we have a system of n point masses m\ with radius vectors r and velocities r. We define a quantity A (the "virial" of the system) as follows:

The time derivative of this is n

The first term equals twice the kinetic energy of the ith particle, and the second term contains a factor mir, which, according to Newton's laws, equals the force applied to the ith particle. Thus we have

Fig. 6.12. When a system consists of more than two bodies, the equations of motion cannot be solved analytically. In the solar system the mutual disturbances of the planets are usually small and can be taken into account as small perturbations in the orbital elements. K.F. Sundman designed a machine to carry out the tedious integration of the perturbation equations. This machine, called the perturbograph, is one of the earliest analogue computers; unfortunately it was never built. Shown is a design for one component that evaluates a certain integral occurring in the equations. (The picture appeared in K.F. Sundman's paper in Festskrift tillegnad Anders Donner in 1915.)

Fig. 6.12. When a system consists of more than two bodies, the equations of motion cannot be solved analytically. In the solar system the mutual disturbances of the planets are usually small and can be taken into account as small perturbations in the orbital elements. K.F. Sundman designed a machine to carry out the tedious integration of the perturbation equations. This machine, called the perturbograph, is one of the earliest analogue computers; unfortunately it was never built. Shown is a design for one component that evaluates a certain integral occurring in the equations. (The picture appeared in K.F. Sundman's paper in Festskrift tillegnad Anders Donner in 1915.)

where T is the total kinetic energy of the system. If {x) denotes the time average of x in the time interval [0, t], we have i = 1

If the system remains bounded, i. e. none of the particles escapes, all ri's as well as all velocities will remain bounded. In such a case, A does not grow without limit, and the integral of the previous equation remains finite. When the time interval becomes longer (t ^ to), {A) approaches zero, and we get

where rij = \ri — rj |. The latter term in the virial theorem is now n n

rJi rij

Since (ri — rj ) • (r — rj ) = r2 the sum reduces to nn

This is the general form of the virial theorem. If the forces are due to mutual gravitation only, they have the expressions

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