## EJSS Earth and Moon Model

## 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:

δ x δ t=v x
δ y δ t=v y
δ z δ t=v z
δ v x δ t=−G M x(x 2+y 2+z 2)1.5
δ v y δ t=−G M y(x 2+y 2+z 2)1.5
δ v z δ t=−G M z(x 2+y 2+z 2)1.5

## The equations of rotation are:

r o t a t i o n e a r t h=r o t a t i o n e a r t h+1 δ t
r o t a t i o n m o o n=r o t a t i o n m o o n+1 360 δ t

## Equations used to calculate physics quantities are:

r=x 2+y 2+z 2−−−−−−−−−−√
v=v 2 x+v 2 y+v 2 z−−−−−−−−−−√
t h e t a=t a n−1 y x
The model used δ t=1, the time taken for Earth to rotate i complete revolution is therefore
t d a y =t 360
so after 360 δ t steps,
t d a y =1
To calculate period T,
o m e g a=v r
therefore,
T=2 π ω
For collision detection, the model assumption used is
r< r E a r t h+r M o o n
For largely visualization purposes,
r E a r t h=0.637 some what familiar to real data
r M o o n=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 E a r t h=0.6=6 x 10 24 k g
G=0.667 k=6.67 x 10−11 m 3 k g−1 s−2
where k=0.58 x 10−4 to achieve comparable period T=27.3 days
r=3.844= 385 000 k m
and velocity of moon is
v c a l=(v)(1 x 10 9)k 1 where k 1=2.4 arbitrarily determined

### Credits

The EJSS Earth and Moon Model was developed by Francisco Esquembre, and Loo Kang WEE using the Easy Java Simulations (EJS) version 5.0 authoring and modeling tool.