Monday, December 30, 2013

EJSS Earth and Moon Model

EJSS Earth and Moon Model by paco remixed by lookang.


http://weelookang.blogspot.sg/2013/12/ejss-earth-and-moon-model.html
https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_model_EarthAndMoon3Dwee/EarthAndMoon3Dwee_Simulation.html
source: https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_src_EarthAndMoon3Dwee.zip
http://weelookang.blogspot.sg/2013/12/ejss-earth-and-moon-model.html
https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_model_EarthAndMoon3Dwee/EarthAndMoon3Dwee_Simulation.html
source: https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_src_EarthAndMoon3Dwee.zip
https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_model_EarthAndMoon3Dwee/EarthAndMoon3Dwee_Simulation.html

Description:


The Moon completes its orbit around the Earth in approximately 27.32 days (a sidereal month). In this model, we assume the Moon to orbit about the center of the Earth. By this assumption, the Moon is at a distance of about 385000 km from the center of the Earth, which corresponds to about 60 Earth radii. With a mean orbital velocity of 1.023 km/s,[1] The Moon orbit is modeled to be a perfect circular motion orbit, a close approximation the real Moon's orbit. The model also assume the Moon to move on the Earth's equatorial plane.

The equations of motion are:



$\frac{\delta x}{\delta t} = v_{x} $

$\frac{\delta y}{\delta t} = v_{y} $

$\frac{\delta z}{\delta t} = v_{z} $

$\frac{\delta v_{x}}{\delta t} = - \frac {GMx}{(x^{2}+y^{2}+z^{2})^{1.5}} $

$\frac{\delta v_{y}}{\delta t} = - \frac {GMy}{(x^{2}+y^{2}+z^{2})^{1.5}} $

$\frac{\delta v_{z}}{\delta t} = - \frac {GMz}{(x^{2}+y^{2}+z^{2})^{1.5}} $

The equations of rotation are:

$\ rotation{earth} = rotation_{earth}+\frac {1}{\delta t} $

$\ rotation{moon} = rotation_{moon}+\frac {1}{360 \delta t} $


Equations used to calculate physics quantities are:


$\ r = \sqrt{x^{2}+y^{2}+z^{2}} $

$\ v = \sqrt{v_{x}^{2}+v_{y}^{2}+v_{z}^{2}} $

$\ theta = tan^{-1} \frac {y}{x} $

The model used $ \delta t = 1$, the time taken for Earth to rotate i complete revolution is therefore
$\ t_{day} = \frac{t}{360} $

so after 360 $ \delta t$ steps, 
$\ t_{day} = 1 $

To calculate period T,  

$\ omega = \frac {v}{r} $

therefore, 

$\ T = \frac {2 \pi}{\omega} $

For collision detection, i used

$\ r <  r_{Earth}+r_{Moon} $

For largely visualization purposes, 
 $ r_{Earth} = 0.637$ some what familiar to real data

$ r_{Moon} = 0.1737 $ not to scale

in order to create realistic simulation, the model used constants to create numeric that corresponds to the real world.

for example in the model versus in the world,
$ M_{Earth} = 0.6 = 6 x10^{24} kg$

$ G = 0.667 k = 6.67 x10^{-11} m^{3} kg^{-1} s^{-2}$

where $ k = 0.58x10^{-4}$ to achieve comparable period $T = 27.3$ days

$ r = 3.844 = 385 000 km $
and velocity of moon is 

$ v_{cal} = \frac {(v)(1x10^{9})}{k1} $ where $k_{1} = 2.4$ arbitrarily determined

changes made:

added realistic numbers in the model
added better graphics from creative commons or public domain pictures
added plane of Moon's rotational plane
added control and sliders to suit my usual design ideas


Activities:

set the velocity to zero and play the simulation, is it as you expected it to observe? What is the physics concepts that can be used to explain this?