Solar system

 

      Kepler's first law

 

  The simulation shows the Earth in an elliptical orbit during its revolution around the Sun.

  Keplers first law says: each planet moves around the Sun in an ellipse, with the Sun on the focus.

 

  Button  allows a choice between a viewpoint perpendicular to the plane of the ecliptic or an axonometric view.

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

 

 

 

 

 

       Kepler's second law

 

  During its revolution the Earth moves at its maximum speed in the perihelion (minimum distance of the Sun) and at its minimum speed in the aphelion (maximum distance). In fact, the radius vector from the Sun to the planets sweeps along equal areas in equal intervals of time. Therefore, as the length of the radius varies in time, the elliptical sector travelled in a unity of time will vary too.

  All this is expressed by saying that the area of elliptical sector swept in a unity of time remains constant.

 

 

 

 

 

       Kepler's third law

 

  Keplers third law says that the squares of the periods of any two planets are proportional to the cubes of the semi-major axes at their respective orbits.

  The simulation shows four planets set at different distances that rotate around a star. A timer beats the time in arbitrary unities to make it possible to measure the periods of the four planets.

  On the status bar we show the respective periods and the respective maximum distances of the four planets from the star.

 

  Button  makes the orbits of the planets visible.

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

 

 

 

 

 

       Relative motion Earth-planet

 

  On the left side of the screen the simulation shows the motion of the Earth and of a planet around the Sun. The circumferences formed with points and with the Sun in the centre represent the vault of the heaven.

  On the right side of the screen it is possible to see the motion of the planet as it is seen from the Earth.

  Because of the different period of revolution, for a certain period of time, we observe an inversion of the motion (retrograde) of the planet in relation to the Earth.

 

  On the tool bar it is possible to choose between a planet inside the terrestrial orbit (Mercury) and an external orbit (Mars).

  In the case of the external planet, by clicking on button , it is possible to draw a line that, starts from the Earth, crosses Mars and continues to the vault of the heaven. This helps to understand the motion of Mars seen from the Earth and in particular the retrograde motion.

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

 

 

 

 

 

       Apparent path of the Sun

 

  During the year the Sun changes its course on the vault of the heavens. The journey depends also on the latitude of the observer.

  The simulation shows the course of the Sun for each month of the year starting from the winter solstice.

  A legend indicates the solstices and the equinoxes.

 

  The choice of the latitude in degrees and primes is made in the tool bar.

  With button  it is possible to keep visible the position occupied by the Sun in correspondence of the equinoxes and the solstices.

  Start by clicking the button .

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

 

 

 

 

 

      Irradiation E

 

  We have a point source of light. A sheet that rotates around its axis is on the path of the rays. The sheet forms a shadow on a screen. The surface of the shadow depends on the surface of the sheet, on its slope in relation with the rays and on its distance from the source.

  The irradiation E is the ratio between the radiant flux that drops normally to a flat surface S and the surface itself. The irradiation depends on the size of the surface (direct proportionality), on the distance between the source and the surface (inverse square proportionality) and on the angle between the surface and the rays. In particular it depends on the cosine of the angle that the unit vector normal to the surface forms with the direction of the rays.

  The shadow on the screen gets larger as the radiant flux intercepted by the sheet gets larger and therefore so does the radiation.

  On the status bar it is possible to see the distance from the source to the sheet and the radiation E.

 

  On the tool bar it is possible to vary the distance of the sheet from the source.

  By clicking the button the unit vector normal to the surface is shown.

  To have a continuous progress click on button .

  To have a step-by-step progress click on button .

  Click the button once the step-by-step progress has been chosen. The progress can also be obtained with the key (Enter).

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

 

 

 

 

 

    Axonometry

 

  It is a graphic representation of spatial figures.

 

 

 

 

 

    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.