Celestial mechanics

 

 

        Satellite motion

 

  The satellite motion is due to forces which are directed towards a point. The gravitational forces are parts of these forces.

  In this simulation we show the variation of the direction and the magnitude of this force acting between a fixed body and a body rotating around an elliptical orbit (Sun Earth). We also show the direction and magnitude of the velocity, of the tangential and centripetal acceleration of the moving body.

 

  Button  shows or hides the force vector.

  Button   shows or hides the velocity vector.

  Button   shows or hides the centripetal acceleration vector.

  Button  shows or hides the tangential acceleration vector.

  Button is used to maintain the vectors temporary or permanently visible.

 

 

 

 

 

         System of two bodies (double stars)

 

  Two bodies, as for example the double stars, rotate one around the other in relation to their centre of mass. Note that the radius of the orbits is greater when the mass is smaller. Because of the fact that the period of revolution is the same, the body with a smaller mass runs with a greater speed.

 

  On the tool bar it is possible to choose the ratio between the masses.

  Button  allows the choice between a point of view perpendicular to the plane of rotation and an axonometric point of view.

  Button  makes the orbits of the planets visible.

  By clicking the button  it is possible to start a new simulation.

 

 

 

 

 

         Precession and nutation

 

  In this simulation we show a schematisation of the precession and the nutation of the Earth.

  The motion of precession is the motion for which the Earth's axis of rotation does not maintain a fixed direction but also rotates around another axis.

  Because of this motion the star which indicates the North Pole varies in time.

  The motion of nutation is a slight oscillation of the Earth's axis of rotation which is superimposed on the precession.

 

  By clicking the button   we can see the motion of the Earths equator.

 

 

 

 

 

          Top

 

  The motion of the top is a motion composed of a rotation around itself and one of its rotation axis around the vertical. It is a precession motion similar to the precession of the Earth.

  The motion can be see from above or sideways.

 

  Buttons   and   allow the choice of the point of view.

 

 

 

 

 

      Model of expansion of the universe

 

  A sphere on which surface circles are drawn expands. The distance between the circles grows.

 

  In the tool bar it is possible to choose between an open or closed universe.

  By clicking on button   it is possible to start a new simulation.

 

 

 

 

    Axonometry

 

    Axonomet

  It is a graphic representation of spatial figures.

 

 

 

 

 

    Force of attracion

 

   This is the force which two masses M and m attract one another. It is proportional to the mass, and inversely proportional to the square of the distance between two centres.

.

  where G is the universal gravitational constant.

 

 

 

 

 

    Centre of mass

 

  Lets consider a system of bodies of mass along a line, at different distances from a point O.

  The centre of mass is the point:

  It is possible to broaden the definition to the space by considering also coordinate y and z.

 

 

 

 

 

    Vector

 

  A vector is a quantity characterized by the fact that has a direction and a magnitude.

  It is represented by an oriented segment in which:

  • The direction is given by the line the segment is part and by the parallels.
  • The magnitude is given by the length of the arrow once a scale has been introduced.

 

 

 

 

 

    Velocity

 

  The speed v is given by the ratio between the travelled space and the time needed to travel it. Very short intervals will be considered so that the ratio does not vary.

  The speed is a vector quantity. It is measured in meters per second (m/s).

 

 

 

 

 

    Tangential acceleration

 

  There is tangential acceleration when the magnitude of the velocity varies. It has the same direction as the velocity.

 

 

 

 

 

    Centripetal acceleration

 

  There is centripetal acceleration when the direction of velocity varies, i.e. when the trajectory is not rectilinear. Its direction is perpendicular to that of the velocity. It is directed towards the centre of curvature of the arc of the trajectory. The magnitude is worth:

 

 

 

 

 

    Proportionality

 

  • Between two quantities a and b there is a direct proportionality when the ratio a/b = k is constant.
  • Between two quantities a and b there is an inverse proportionality when the product a * b = k is constant.
  • Between two quantities a and b there is square proportionality when the ratio between one of them and the square of the other a /b˛ = k is constant.
  • Between two quantity a and b there is an inverse square proportionality when the product between one of them and the square of the other a * b˛ = k is constant.