kill(all)$

load(lrats)$

simplified approach thrid law of kepler( =  use circular orbit ),version 1.0
peter.vlasschaert@gmail.com,03/08/2018

We use equation , polar equations  ' : 'p30a,p30b'
see :rocketlaunchfreeflightversion1final.wxmx  ('see website')
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p1:('diff(r,t,2)-r*('diff(θ,t,1))^2)=-(G*M_e)/r^2;
p2: r^2*('diff(θ,t,1))=V*r_0*cos(γ);

h=cte ,when orbit stays one plane.
rem :motionstaysintheplane..wxmx  ( ' see website ')

p3:p1;
p4:lhs(p2) = h;

' p3' = ' force balance to stay in orbit around earth : Me ' for 'm'

When use circular orbit : r = R = constant
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rem : r=r(t) = R , ω = 'diff(θ,t) for 'circle',  into  'p3,p4'
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p5:[r=R,'diff(θ,t)=ω];
p5a:subst(p5,p3);
p5b:subst(p5,p4);

p6a:ev(p5a,diff);
p6b:ev(p5b,diff);

rem : centripetal force = gravitational force  ' = p6a'
==========================================
rem : centrifugal force = - centralpetal force.
=>  Why 'm' stays in orbit

ω = 2*π*f = 2*π*1/Τ
================
f : frequency
T : period
================
relation =>  f = 1/T
-----------------------------

p7:[ω =2*π*1/T];
p7a:subst(p7,p6a);
p7b:subst(p7,p6b);

To  show thrid law kepler :  use 'p7a'
==============================
=>  circular orbit 

p8:p7a*R^2;

coef :R^3/T^2
============

p8a:R^3/T^2$

p8b:ratcoeff(lhs(p8),p8a);

simplified 'p8'
===========

p8c:p8/p8b;

assume (G>0,M[e]>0);

rem : rhs(p8c) = C1 = const
=======================

p8d:lhs(p8c) = C1;

p8e:1/p8d;

rem : C1 =const1 , 1/C1 = const2

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p9:lratsubst([1/C1=const2],p8e);

simplified approach 3e law kepler  'p9'
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M[e] = M[sun], m = planet .

The squares of the periods = T of the planets are proportional to
the cubes of the radius = R.

elliptical orbit :  R=a  , a = semi-major axis.
===================================

p10:subst([R=a,const2=const],p9);

general thrid law kepler
====================
M[e] = M[sun], m = planet .

The squares of the periods = T of the planets are proportional to 
the cubes of their semi-major axis  = a.

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