Vis-viva equation

The celestial mechanics Vis -Viva equation provides the local velocity of bodies in Keplerian orbits around a dominant celestial body. Under these conditions, the sum of the rate- dependent kinetic energy - that's half the historical Vis viva - and the distance-dependent energy in the gravitational field constant in time ( conservation of energy ). For a given gravitational parameter different values ​​of this sum yield different track shapes. Of the orbital parameters, only the semi-major axis is included in the equation.

Vis -Viva equation

The Vis Viva equation for the instantaneous velocity of a body in an elliptical (whether the circle here as a special case of the ellipse miteinbegriffen ), parabolic or hyperbolic orbit around a central star is located, is:

Here, the distance of the body from the center of gravity, the square of its speed, the semi-major axis of the cone section in circulation ( for a circle, an ellipse, a parabola and a hyperbola ), and the standard gravitational parameter as gravitation constant and the mass of the central body:

Taking the square root of the above expression for, is obtained as the speed of the orbiting body:

A circular path is obtained by employing the following simplified relationship can also be obtained by equating the gravitational and centripetal force:

This special rate is also called minimum circular velocity or first cosmic speed: moves a circumferential body at that speed around the central body, so its orbit is a circle. Is the rotational speed on the other hand (at constant distance ) is less than or greater than, is formed as an elliptical orbit.

If the semi-major axis infinitely large, arises as a "degenerate " ellipse with a second focal point at infinity a parabola. For such parabolic paths are obtained accordingly by inserting a simplified form of the equation:

This special rate is also called escape velocity or second cosmic speed: moves a circumferential body with this or a higher speed, so that he can overcome the gravitational binding to the central body and its orbit is no longer closed but open. If the velocity of circulation of the body it exactly the same, the orbit is a parabola, with larger orbital velocities on the other hand ( in otherwise constant distance ) a hyperbola.

Is not negligible, the mass of the orbiting body relative to the mass of the central body, one can no longer assume that the center of gravity ( center of mass ) of the system lies in the center of the central body. The mass of the circular ends of the body ( the so just not a " specimen " is more ) then has to be taken into account; whereby the Vis -Viva equation changes as follows:

The speed so described is the speed at which the particular body in its orbit for an outside observer, or in respect to a given fixed point ( eg the center of mass ) moves.

The relative speed of the two bodies with respect to each other then the velocity of the specimen corresponding to the central body, which is described with the conventional Vis Viva equation (1).

Derivation

For the derivation of the Vis -Viva equation there are different approaches or approaches. The following three options are shown which relate specifically to elliptical orbits and therefore the decisive step to the relations or and draw the resultant from the second Kepler 's law equation, where the semi-major axis of the elliptical orbit, and their peri-and Apoapsisdistanz as well and the associated web speeds are.

The derivations are performed under the assumption that the mass of the rotating body in relation to the central body is negligible (1). If you want the mass of the orbiting body takes into account ( 2), so must the reduced mass to be used in place of normal body mass in the kinetic energy:

This is a result of the fact that the total kinetic energy of the two bodies concerned is the sum of the kinetic energies:

Is relative to the inertial frame of the barycenter, the corresponding relative velocity of the rotating body and the relative speed of the two bodies to one another, the speed of the central body.

First option

According to the conservation of energy in the gravitational field of a central mass is the sum of potential energy and kinetic energy of a body of mass constant.

Caused by the mass of the central body, gravity force acting on a body of the composition is according to the following formula based on the distance of the body from the center of gravity of the central body, in the case of a homogeneous sphere of its center so:

The potential energy gained by the body when it is brought from the center of gravity of the surface of the central body up to a position at a distance, results therefore from the integration of the force acting on it gravitational force along the from it while the distance covered, wherein there, the gravitational force acts radially, the increase of the potential energy alone from the conquered height difference depends, any sideways movement here so no role to play:

It is the output radius of the instantaneous distance of the body from the center of gravity, is integrated over, and reached the end of the body away from the center of gravity.

If we want to use from now on the gravitational parameter to simplify the notation, instead of the product, provides the above integration as a result of finally following expression:

The kinetic energy of the body is known as:

The energy conservation law for a moving in the gravitational field of a central mass moving body says that the sum of its kinetic and potential energy along the trajectory remains constant, the total energy of the body so it can be formulated more generally as follows:

Now we consider the total energy of the two arbitrary points and with:

Division by and subtraction of supplies on both sides of the equation:

Changed by then results:

If we transfer this equation first of all applicable to any two points in space on an ellipse, we can and for as well and also, for example, the velocities and the Apozentrum and pericentre and use the apoapsis and periapsis distance and:

With the aid of relations or and the opportunities arising from the second Kepler 's law and conservation of angular momentum equation can be obtained for just the formula once again be simplified as follows:

We now replace in the equation for the point by the "real" arbitrary ellipse point without all the indexes and the parameters of the second point by the Apozentrums, we obtain the following equation which can be easily simplify the sought- Vis Viva equation:

Second possibility

The gravity or gravitational force of a mass whose center is located at a distance from the center of a second mass can be calculated using the law of gravity as follows:

Considering a body whose mass is negligible relative to the mass of the central star, as is the potential energy of the body that work is what is done against the gravitational force, when this body suspended from a point at a distance from the central body to infinity will.

Thus, its potential energy is calculated with:

If this factor is replaced by the gravitational parameters, the integration yields the expression:

The kinetic energy of the body is known as:

The energy conservation law for a moving in the gravitational field of a central mass moving body says that the sum of its kinetic and potential energy along the trajectory remains constant, the total energy of the body so it can be formulated more generally as follows:

If we transfer this equation to an ellipse and use the speed in Apozentrum and the apoapsis distance for and, for example, we obtain the following relationship:

Similarly, we obtain for the pericentre the two mutually equivalent equations:

With the aid of results from the second Kepler 's law and conservation of angular momentum equation can be the formula just obtained for a for transform and the expression obtained subsequently used in the above energy equation of Apozentrums that are or by changing gradually the aid of the relations again can be simplified:

Inserting the obtained expression for the general equation of the total energy

And after rearranging yields in this case finally the Vis -Viva equation:

Third option

The third possibility is the same again from the two previously unknown total energies in the apo - and pericentre, but this then multiplied by the square of the respective path speed and:

Again with the help of the previously mentioned equation, we can now replace in the second of the two new equations obtained by the expression, and then take the difference of the two equations:

Division through and replace the resulting denominator by supplies as before the location-independent energy

And from this on the general equation of the total energy

The sought- Vis Viva equation:

Example: web speeds in the solar system

In the solar system, the sun is the dominant central body. The mass of the Earth is, for example, only 1/330.000 mass of the Sun and can be neglected in the application of Vis Viva equation - the error is smaller than the neglect of perturbations by Jupiter. With neglected m is constant, and it is close to pull this constant up to a length unit from the root and calculate the prefactor.

Distances in the solar system are often in astronomical units. So we pull out from the root. The pre-factor has then not random (see Gaussian gravitational constant ) the value

The mean track speed of the earth around the sun.

We will first calculate the velocity of the Earth also at perihelion and aphelion. The distances from the sun on these two points by train be 0.983 or 1.017 AE AE and a is, by definition, 1 AU. so

Now the speed of the currently visited by the Rosetta spacecraft comet Churyumov- Gerasimenko at perihelion, aphelion and 3 AU distance. a is 3.503 AU.