What is the spiral shape produced by an attractive (1/r^2)sin(theta) force?

[kmarinas86]

I was wondering what shape a (1/r^2)sin(theta) force would produce between two bodies.

theta = angle between the distance between the two bodies and the velocity vector (90 degrees for a circular orbit).

 

[kmarinas86]

Here I broke it up into pieces:

r = distance from the origin

s = sin(atan(r*θ' / r'))

(A = constant)

r'' = A*s/r^2

System of equations:

r'' = A*s/r^2

r = (A*s/(r''))^0.5

I have no idea how to plot this. I know only as much as the fact that I can set initial values, but I don't know how to get from there to actually plotting the graph.

 

[JJacquelin]

s = sin(atan(r*θ' / r'))
(A = constant)
r'' = A*s/r^2
System of equations:
r'' = A*s/r^2
r = (A*s/(r''))^0.5

r'' = A*s/r^2 and r = (A*s/(r''))^0.5 is the same equation.
So, you have one equation only. Eliminating s leads to :
r'' = A*sin(atan(r*θ' / r'))/r^2
Since the equation contains two unknows (r ans theta), it is not solvable.
The physical modeling is unfinished.