Category Archives: Uncategorized

wxWidgets for gnuplot 4.6.0 (ubuntu 12.04)

  • libwxgtk2.8-dev – for the wxt terminal
  • libpango1.0-dev – for the cairo (pdf, png) and wxt terminals
  • libreadline5-dev – readline support (editing command lines)
  • libx11-dev and libxt-dev – X11 terminal
  • texinfo (optional) – needed for the tutorial
  • libgd2-xpm-dev (optional) – old png, jpeg and gif terminals based on libgd

it took some minutes for my stupid head to find out what was missing 😛

( actually  ”Package ‘libreadline5-dev’ has no installation candidate” )

it’s better to install one-by-one these packages .

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Latex

so typing latex is always cool , after the great community of mathlinks it’s time to remeber some of good-old type Latex here at wordpress .

There are many tutorials for learing latex but an easy way is also using mathurl.com , exporting your desirable math expression , exporting and a png and having the code too .
For more latex info see this nice document :

Click to access LaTeX_Manual_8_6.pdf

For linux users install it with the following commands :
$ sudo apt-get install texlive
(a basic subset of TeX Live’s functionality)
$ sudo apt-get install texlive-full
(all the packages in the LaTeX distribution )
$ sudo apt-get install gedit-latex-plugin
(usefull plugin to convert Gedit into a LaTeX editor )

LATEX ART

type this :
$ \setlength{\unitlength}{2mm}\begin{picture}(30, 20)\linethickness{0.075mm}\multiput(0, 0)(1, 0){31}{\line(0, 1){20}}\multiput(0, 0)(0, 1){21}{\line(1, 0){30}}\linethickness{0.15mm}\multiput(0, 0)(5, 0){7}{\line(0, 1){20}}\multiput(0, 0)(0, 5){5}{\line(1, 0){30}}\linethickness{0.3mm}\multiput(5, 0)(10, 0){3}{\line(0, 1){20}}\multiput(0, 5)(0, 10){2}{\line(1, 0){30}}\end{picture} $

and we get :

\setlength{\unitlength}{2mm}\begin{picture}(30, 20)\linethickness{0.075mm}\multiput(0, 0)(1, 0){31}{\line(0, 1){20}}\multiput(0, 0)(0, 1){21}{\line(1, 0){30}}\linethickness{0.15mm}\multiput(0, 0)(5, 0){7}{\line(0, 1){20}}\multiput(0, 0)(0, 5){5}{\line(1, 0){30}}\linethickness{0.3mm}\multiput(5, 0)(10, 0){3}{\line(0, 1){20}}\multiput(0, 5)(0, 10){2}{\line(1, 0){30}}\end{picture}

or draw a circle :

\setlength{\unitlength}{.5cm}\begin{picture}(2,2)\qbezier(2,1)(2,2)(1,2)\qbezier(1,2)(0,2)(0,1)\qbezier(0,1)(0,0)(1,0)\qbezier(1,0)(2,0)(2,1)\end{picture}

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Chaotic motion of solar system

it ‘s been a while since i saw , this question at a  postgraduate cfd course .

Math model

Mathematical description of the basic laws of motion  has to do with the universal law of gravitation. Thus, we came to a simple set of equations that determine the motions of the planets. The force on each planet is simply the sum of the gravitational forces from the sun and all of the other planets in the solar system. In vector notation, this is expressed as:

 

                     m_{i}\frac{d^{2}x}{dt^{2}} = Gm_{i}\sum_{i.ne.j}m_{j}\frac{(r_{j}-r_{i})}{(r_{j}- r_{i})^3}      ,

  where G is the universal constant of gravitation, m values are the masses, and r values are their positions in space.For 2 masses it's easy to see that the solution is a conic . For more body-system it's more complex.

Gravitational resonances

 Chaos in the solar system is associated with gravitational resonances. The simplest case of a gravitational resonance occurs when the orbital periods of two planets are in the ratio of two small integers, e.g., 1:2, 3:5, etc. There are other more subtle gravitational resonances when one considers the precessional periods of planetary orbits in addition to their orbital periods. Strong and weak resonances thread the entire phase space of the solar system in a complex web. Overlapping resonances , i.e., multiple gravitational resonances in close proximity, provide the route to chaos in the solar system. Gravitational resonances may effect very large orbital changes or only modest orbital changes (in some cases, even provide protection from large perturbations), depending sensitively on initial conditions. The long term dynamics of the planetary system is the dynamics of gravitational resonances.

Orbits ...

 Although the numerical simulations all indicate chaos in planetary orbits, in a qualitative sense the planetary orbits are stable—because the planets remain near their present orbits—over the lifetime of the sun. However, the presence of chaos implies that there is a finite limit to how accurately the positions of the planets can be predicted over long times. Of all of the planets, Mercury's orbit appears to exhibit changes of the largest magnitude in orbital eccentricity and inclination. Fortunately, this is not fatal to the global stability of the whole planetary system owing to the small mass of Mercury. Changes in the orbit of the Earth, which can have potentially large effects on its surface climate system through solar insolation variation, are found also to be small.

 The orbital motion of the planets in the Solar System is chaotic. After Pluto ,the evidence that the motion of the whole Solar ,system is chaotic was established with the averaged equations of motion and confirmed later on by direct numerical integration . The most immediate expression of this chaotic behavior is the exponential divergence of trajectories with close initial conditions.

Indeed, the distance of two planetary solutions, starting in the phase space with

a distance d(0) = d_{0} , evolves approximately as  d(T) = d(0)e^{\frac{T}{5}} or in a way which

is even closer to the true value,  d(T) = d(0)10^{\frac{t}{10}}

The amount of time which we allow us see , chaos-in-action ( i mean the effects of chaos ) is huge , so we will continue to estimate the possition of the moon , the other planets and comets with satisfactory precession despite chaos ( for axample if an asteroid is gonig towards earth in the next 1000 years ) , but if want to compute the position of the solar system after some hundreds-of billions years we will propably fail .

I 'm not familiar with the field of astronomy , i maybe forgot something or made a mistake , any comments will be great ...

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