Solving the Three Body Problem for Proton Decay Trajectory

[m_annapaula]

Hello, I'm trying to simulate the trajectory of the decay of two protons from the nucleus. I am using classical physics to calculate the trajectory from Newton's second law. For this, I wrote a python program, but I'm not getting the expected result, which would be an exponential path.
My analysis is part of the principle that protons are embedded in the nucleus but still "stuck" to it. Place the reference system with (0,0) at this point. Stipulating the protons upwards (positive part of the axis) and one down (negative part of the axis), I am analyzing only the positive proton and the nucleus.
Here are the initial conditions and the equations I am using, where r = proton position vector and R = nucleus position vector. The proton starts at the point (R, d / 2) and the nucleus at (0,0) and both the initial velocities are zero. Initial conditions:

R = 5.0 fermi #Ray of an Iron Atom
d = 1.0 fermi #Distance between protons
Q = 26 C #Number of protons
q = 1.0 C #Charge of a proton
k = 1.44 MeV.fm #Electrostatic Constant
m = 931 MeV #Mass of a Proton
M = 45 * m #Mass of iron's nucleus
t = 100 fermi/c #Total time
dt = 1 fermi/c

Equations for proton:

Ax = K*q*Q*x / | r - R |**3
Ay = k*q*Q*y / | r - R |**3 + k*q^2/4*y^2

Vx = Vox + Ax*dt
Vy = Voy + Ay*dt

X = Xo + Vox*dt + (Ax*dt^2)/2
Y = Yo + Voy*dt + (Ay*dt^2)/2

Equations for nucleus:

Ax = 2*(m/M)*Ax(proton aceleration)

Vx = 2*(m/M)*Vx(proton velocity)

X = Xo + Vx*dt + (Ax*dt^2)/2

The expression for acceleration and core velocity I find through the conservation of the linear momentum and then derive the equation to find the acceleration.
It may not be clear enough, but I tried to make it as clear as possible. The equations seem to be correct, but the trajectory I get is a straight line, not an exponential one. Can you help by looking at the physical part?

 

[BvU]

Hello m_annapaula, $\qquad$ $\qquad$ !

Interesting exercise. Some start-off remarks:

• I never heard of such a decay
• The way you describe your scenario it's more a two-body problem (good!)
• Why make things difficult from the onset, instead of doing a one body problem first to check

$\qquad$ $\qquad$ (i.e. fix nucleus at (0,0) a let a proton fly off along the x-axis)

And now my two cents to help you further:

Could you explain your system of units ? If I just fill in the numerical values for the symbols, I get a very small acceleration, so very little happens in the first 100 steps ...

I miss the m in Ax

Vx = Vox + Ax*dt

This isn't right. You want something like $v_x(r_{n+1}) = v_x(r_n) + A_x(r_n) \cdot {\mathrm d}t$.

Similarly X = Xo + Vox*dt + (Ax*dt^2)/2 : it's a formula for linear motion with constant acceleration and all you do is make Ax vary.