Response of Test Particle in a Potential
Subjected to an External Force Pulse
Joachim Köppen Kiel Nov 2017

Some brief explanations

Some insight in the mechanism of the removal of gas from the disk of a spiral galaxy by ram pressure stripping can be gained by considering the movement of stars or parcels of gas perpendicular to the galactic plane, which follows the perturbation by a pulse of an external force. This is explored in the simulation.

Consider a parcel of gas at some galactocentric radius, and let the parcel be represented by a test particle (a mass point) which sits in the gravitational potential caused by the stars and gas of the galaxy. Let us regard only the motion perpendicular to the galactic plane. Then the potential as a function of height z above the plane has the characteristic form of a well of some finite depth, for instance Φ(z) = - vesc²/2 /(1+z²). It is convenient to use normalized units: Let us set vesc = 1 so that all velocities are measured in terms of the escape speed from the galactic plane. Furthermore, z is given in terms of the scaleheight of the vertical potential.

Initially, the particle is at rest in the galactic plane z=0. It shall have unity mass. For small deviations from the rest position, the particle will experience a restoring force, which increases linearly with the deviation. If it is given an initial velocity vz(0) or it is perturbed, it will perform small amplitude oscillations around its rest position. The length of one period is in the normalized units.

The action of the ram pressure is described by a pulse of an additional force with specified strength and duration: F(t) = F0 f(t). Then the particle will move according to its equation of motion: d²z/dt² = Φ(z) + F(t)
This equation is solved numerically using a 4th order symplectic method. Thus the momentum is conserved with machine accuracy.

The controls of the simulator are:

In the solutions one may distinguish two limiting cases:

Pulse of external force

peak force F0

Initial conditions of the particle
position z
velocity vz [vesc]

time of escape
position z of escape

available momentum
received momentum
maximum time
time step

X-axis =
Set range: ...
Y-axis =
Set range: ...

Mouse position:

Sample plots

Blue curves indicate that the particle is bound in the potential, red curves show that it has reached positive total energy and is escaping. A green curve, labelled by E=0, indicates the locus of zero total energy. Magenta lines mark the height zmax where the restoring force has its maximum value.
The Long Pulse Limit: reponse to a constant external force:

For the parabolic potential well the restoring force has its maximum value of 0.3247 at a height zmax=0.577, where the potential energy is still negative -0.375. Thus, if the particle is pushed with a force equal to or greater than this maximum value, the particle can be brought up well beyond that height but it remains bound in the well. Thus more work has to be done by the force to liberate the particle from the potential.

If we push the particle with a force of 0.324, the value of the maximum restoring force, it reaches that height zmax with a speed of 0.347, but its kinetic energy of 0.12 is still insufficient to overcome the potential energy. Since the force continues at a constant level, the particle achieves escape speed at time = 4.78 at a height of 1.55, well above the height for maximum restoring force.

However, the particle can already be detached by a somewhat lower force: the minimum force necessary for the particle's escape is 0.25. Here, the particle passes zmax with a speed of 0.19, which decreases to nearly zero at z=1. If it can pass this critical point, it will eventually be reaccelerated to escape speed at a height of 2 at time=20.

If the force level is much higher, escape occurs earlier and at lower heights. For F=0.9 this happens at the height of maximum restoring force at time 1.2, i.e. one sixth of the period for vertical oscillations.
The Short Pulse Limit: reponse to a short and weak external force pulse:

The particle sits in the parabolic potential well, where the restoring force attains its maximum value of 0.324 at a height zmax=0.56. A Gaussian pulse of force with a time scale of 0.1 (i.e. FWHM=0.17) is applied.

As long as the peak force of the very short pulse does not exceed 5.7, it kicks the particle into oscillations about its rest position at the bottom of the potential well. For a peak force of less than about 2, the amplitude is smaller than the height of the maximum restoring force. With a larger force the particle can be kicked into slow oscillations with amplitudes much higher than that height.

| Top of the Page | back to RPShome | Java Applets Index | JavaScript Index | my HomePage |