We begin with the simplest situation,a sun and a single planet,and investigate a few of the properties of this model solar system.

solar system.gif

According to Newton's law of gravitation the magnitude of the force is given by

![](http://latex.codecogs.com/png.latex?F_G=\frac{G M_S M_E}{r^2})

and we can obtain that:

![](http://latex.codecogs.com/png.latex?\frac{dv_x}{dt}=-\frac{GM_s M_E x}{r^3})

![]( http://latex.codecogs.com/png.latex?\frac{dv_y}{dt}=-\frac{GM_s y}{r^3})

and if we use astronomical units ,AU; and measure time in years, we find

![](http://latex.codecogs.com/png.latex?G M_S=v^2 r=4 \pi^2 AU^2/yr2)

we next convert the equations of motion into difference equations in preparation for constructing a computational solution.We find

![](http://latex.codecogs.com/png.latex?v_{x,i+1}=v_{x,i}-\frac{4\pi^2 x_i}{r_i^3}\Delta t)
![](http://latex.codecogs.com/png.latex?x_{i+1}=x_i +v_{x,i+1}\Delta t)
![](http://latex.codecogs.com/png.latex?v_{y,i+1}=v_{y,i}-\frac{4\pi^2 y_i}{r_i^3}\Delta t)
![](http://latex.codecogs.com/png.latex?y_{i+1}=y_i+v_{y,i+1}\Delta t)

and I imitate it by python ,and I gained that:

Earth Orbiting the Sun

code1，as follows：

#coding:utf-8
import pylab as pl
import numpy as np
import math
from mpl_toolkits.mplot3d import Axes3D
import matplotlib.pyplot as plt
from matplotlib import animation

class circle():
    def __init__(self,x0=1,y0=0,t0=0,vx0=0,vy0=2*math.pi,dt0=0.001,total_time=10):
        self.x=[x0]
        self.y=[y0]
        self.vx=[vx0]
        self.vy=[vy0]
        self.R=x0**2+y0**2
        self.t=[t0]
        self.dt=dt0
        self.T=total_time
    def run(self):
        for i in range(int(self.T/self.dt)):
            vx=self.vx[-1]-(4*math.pi**2*self.x[-1]/self.R**2)*self.dt
            vy=self.vy[-1]-(4*math.pi**2*self.y[-1]/self.R**2)*self.dt
            self.vx.append(vx)
            self.vy.append(vy)
            self.x.append(self.vx[-1] * self.dt + self.x[-1])
            self.y.append(self.vy[-1] * self.dt + self.y[-1])
    def show(self):
        pl.plot(self.x, self.y, '-', label='tra')
        pl.xlabel('x(AU)')
        pl.ylabel('y(AU)')
        pl.title('Earth orbiting the Sun')
        pl.xlim(-1.2,1.2)
        pl.ylim(-1.2,1.2)
        pl.axis('equal')
        pl.show()
a=circle()
a.run()
a.show()

we can use the animation of matplotlib to gain the cartoon,

add follow codes:

 def drawtrajectory(self):
        fig=plt.figure()
        ax = plt.axes(title=('Earth orbiting the Sun'),
                      aspect='equal', autoscale_on=False,
                      xlim=(-1.1, 1.1), ylim=(-1.1, 1.1),
                      xlabel=('x'),ylabel=('y'))
        line=ax.plot([],[],'b')
        point=ax.plot([],[],'ro',markersize=10)
        images=[]
        def init():
            line=ax.plot([],[],'b',markersize=8)
            point=ax.plot([],[],'ro',markersize=10)
            return line,point
        def anmi(i):
            ax.clear()
            line=ax.plot(self.x[0:10*i],self.y[0:10*i],'b',markersize=8)
            point=ax.plot(self.x[10*i-1:10*i],self.y[10*i-1:10*i],'ro',markersize=10)
            return line,point
        anmi=animation.FuncAnimation(fig,anmi,init_func=init,frames=10000,interval=1,
                                     blit=False,repeat=False)

we get follow gif

Earth Orbiting the Sun

If we consider the reduced mass

![](http://latex.codecogs.com/png.latex?\mu\equiv \frac{m1m2}{m1+m2})

The orbital trajectory for a body of reduced mass is given in polar coordinates by

![](http://latex.codecogs.com/png.latex?\frac{d^2}{dt ^2} (\frac{1}{r})+\frac{1}{r}=-\frac{\mu r^2}{L2} F(r))
consider

we have
![]( http://latex.codecogs.com/png.latex?r=(\frac{L^2}{\mu G M_s M_P} )\frac{1}{1-e cos\theta })
so

Then let us suppose that the gravitational force is of the form

![](http://latex.codecogs.com/png.latex?F_G=\frac{GM_S M_E}{r^{\beta}})
then I get
![](http://latex.codecogs.com/png.latex?v_{x,i+1}=v_{x,i}-\frac{4\pi^2 x_i}{r_i^{\beta+1}}\Delta t)
![](http://latex.codecogs.com/png.latex?x_{i+1}=x_i +v_{x,i+1}\Delta t)
![](http://latex.codecogs.com/png.latex?v_{y,i+1}=v_{y,i}-\frac{4\pi^2 y_i}{r_i^{\beta+1}}\Delta t)
![](http://latex.codecogs.com/png.latex?y_{i+1}=y_i+v_{y,i+1}\Delta t)

the picture

Beta=3.0 t=0.3yr v=1.7pi.png

Beta=2.5，t=1.5yr,v=1.7pi.png

Beta=2.3,t=10yr,v=1.7pi.png

code

#coding:utf-8
import pylab as pl
import numpy as np
import math
from mpl_toolkits.mplot3d import Axes3D
import matplotlib.pyplot as plt
from matplotlib import animation

class circle():
    def __init__(self,x0=1,y0=0,t0=0,vx0=0,vy0=1.7*math.pi,dt0=0.001,Beta=2.3,total_time=10):
        self.x=[x0]
        self.y=[y0]
        self.vx=[vx0]
        self.vy=[vy0]
        self.t=[t0]
        self.dt=dt0
        self.T=total_time
        self.beta=Beta
    def run(self):
        for i in range(int(self.T/self.dt)):
            R=(self.x[-1]**2+self.y[-1]**2)**0.5
            vx=self.vx[-1]-(4*math.pi**2*self.x[-1]/R**(self.beta+1))*self.dt
            vy=self.vy[-1]-(4*math.pi**2*self.y[-1]/R**(self.beta+1))*self.dt
            self.vx.append(vx)
            self.vy.append(vy)
            self.x.append(self.vx[-1] * self.dt + self.x[-1])
            self.y.append(self.vy[-1] * self.dt + self.y[-1])
    def show(self):
        pl.plot(self.x, self.y, '-', label='tra')
        pl.xlabel('x(AU)')
        pl.ylabel('y(AU)')
        pl.title('Earth orbiting the Sun')
        pl.xlim(-1,1)
        pl.ylim(-1,1)
        pl.axis('equal')
        pl.show()
    def drawtrajectory(self):
        fig=plt.figure()
        ax = plt.axes(title=('Earth orbiting the Sun '),
                      aspect='equal', autoscale_on=False,
                      xlim=(-1.1, 1.1), ylim=(-1.1, 1.1),
                      xlabel=('x'),ylabel=('y'))
        line=ax.plot([],[],'b')
        point=ax.plot([],[],'ro',markersize=10)
        images=[]
        def init():
            line=ax.plot([],[],'b',markersize=8)
            point=ax.plot([],[],'ro',markersize=10)
            return line,point
        def anmi(i):
            ax.clear()
            line=ax.plot(self.x[0:10*i],self.y[0:10*i],'b',markersize=8)
            point=ax.plot(self.x[10*i-1:10*i],self.y[10*i-1:10*i],'ro',markersize=10)
            return line,point
        anmi=animation.FuncAnimation(fig,anmi,init_func=init,frames=100000,interval=1,
                                     blit=False,repeat=False)
        plt.show()


a=circle()
a.run()
a.show()
#a.drawtrajectory()

then we get the animation

Beta=3.0,v=2.0pi,t=200yr

for the problem 4.8， I use the follow code to calculate

import pylab as pl
import numpy as np
import math
class circle():
    def __init__(self,x0=0.72,y0=0,t0=0,vx0=0,dt0=0.001,Beta=2.0,total_time=10,e0=0.007):
        self.x=[x0]
        self.y=[y0]
        self.vx=[vx0]
        self.vy=[]
        self.t=[t0]
        self.dt=dt0
        self.T=total_time
        self.beta=Beta
        self.e=e0

    def run(self):
        vy0=2*math.pi*(1-self.e)/math.sqrt(1+self.e)
        self.vy.append(vy0)
        for i in range(int(self.T/self.dt)):
            R=(self.x[-1]**2+self.y[-1]**2)**0.5
            vx=self.vx[-1]-(4*math.pi**2*self.x[-1]/R**(self.beta+1))*self.dt
            vy=self.vy[-1]-(4*math.pi**2*self.y[-1]/R**(self.beta+1))*self.dt
            self.vx.append(vx)
            self.vy.append(vy)
            self.x.append(self.vx[-1] * self.dt + self.x[-1])
            self.y.append(self.vy[-1] * self.dt + self.y[-1])
            self.t.append(self.t[-1]+self.dt)
            if(self.y[-1]<0):
                a=(self.x[0]-self.x[-1])/2
                T=2*self.t[-1]
                k=T**2/a**3
                break
        print(k)
a=circle()
a.run()

Exercise_10: The Solar System