Java Trajectories

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The interactive Java applet above allows the dynamics of a trajectory encompassing both radial and orbital velocity components to be visualised. The angular momentum [L] as represented on the applet is normalised in terms of geometric units. Therefore, if you want to convert angular momentum back into units used in previous discussions, the following conversion is required:

[1]      1

Note, the applet assumes the black hole mass to be 1 solar mass [M=1.98E30kg] in line with the previous discussions. However, the initial starting point of the applet positions the mass [m] at the inner radius, which is displayed as a ratio with respect to the Schwarzschild radius [Rs] and corresponds to the angular momentum [N=3.57] as shown.

[2]      2

However, as can be seen, the position at the inner radius corresponds to an unstable knife-edge orbit, which is shown toppling into the energy well of the effective potential. As the mass [m] falls to the minimum of the effective potential curve, the radius [r] increases and, in so doing, it loses kinetic energy in the form of its velocity [v], which now comprises of both radial and orbital components. Given that energy and angular momentum are both assumed to be conserved in this closed system, the loss of kinetic energy is compensated by an increase in gravitational potential, as shown its the position on the grey curve. However, the effects on kinetic energy can probably be best visualised on the right of the applet animation, i.e. the mass [m] can be seen to slow down as the radius [r] increases and vice versa as the radius [r] decreases. However, when the mass [m] reaches the minimum effective potential, at what might now be seen to be confusingly called the outer radius [Ro], it has acquire sufficient radial momentum that the trajectory extends to an even larger radii. Of course, in so doing, mass [m] loses kinetic energy and gains gravitational potential and eventually reaches some maximum radial distance [Rmax], after which the gravitational potential energy exceeds the kinetic energy and mass [m] falls back to some lower radial distance [Rmin]. In this process, the applet shows that mass [m] precesses around the larger mass [M], although in other examples, the mass [m] can either escape the gravitational pull of mass [M] or plunge into the implied black hole.

Can we use the applet to visualise the previous graphs?

If you change the angular momentum on the applet to [4.00], the values of the inner and outer radii will change and align to the first graph under the heading Implications of Effective Potential’. 

[3]      3

However, to do a direct comparison to this previous discussion of circular orbits, you will also have to set the position of mass [m] at the outer radius [Ro]. This can be done by simply clicking on the green vertical line on the applet, which marks the outer radius on the effective potential curve. The following table highlights the data used and the applet should now allow you to seen that a stable circular orbit is indeed possible at [r/Rs=6], but you will be hard pressed to repeat this stability on the knife-edge orbit at [r/Rs=2]:

r/Rs v L=mvr
2.00 2.99E+08 1.77E+12
3.00 1.73E+08 1.53E+12
4.00 1.34E+08 1.58E+12
5.00 1.13E+08 1.67E+12
6.00 9.97E+07 1.77E+12

However, the table above also suggest another special case, for this specific configuration, where [L=1.53E12] is a minimum. This value can again be converted to a value [3.4642] as follows:

[4]      4

The resulting plot of effective potential only has 1 possible circular orbit as the inner and outer values of radius have converged to the same value [r/Rs=3]. Again, if you click of the green line, it is possible to get a stable orbit. However, if you deviate too far either way from this point, the mass [m] will eventually fall into the black hole. In essence, you should be able to re-create the scenarios previously suggested by the following diagram.